College Algebra Exam 3 Study Guide – Step-by-Step Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1a. Simplify: (4 + 3i) − (−7 + 5i). Write your answer in the form a + bi."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Complex Numbers – Addition and Subtraction"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to add and subtract complex numbers, combining real and imaginary parts."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Complex number: "},{"type":"inlineMath","attrs":{"latex":"a + bi"}},{"type":"text","text":", where "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":" is the real part and "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" is the imaginary part."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the subtraction as addition: "},{"type":"inlineMath","attrs":{"latex":"(4 + 3i) - ( -7 + 5i ) = (4 + 3i) + (7 - 5i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the real parts: "},{"type":"inlineMath","attrs":{"latex":"4 + 7"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine the imaginary parts: "},{"type":"inlineMath","attrs":{"latex":"3i - 5i"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1b. Simplify: (3 − 5i)(8 − 2i). Write your answer in the form a + bi."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Complex Numbers – Multiplication"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to multiply two complex numbers using the distributive property (FOIL method)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"FOIL: First, Outside, Inside, Last for multiplying binomials."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remember: "},{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the distributive property: "},{"type":"inlineMath","attrs":{"latex":"(3 - 5i)(8 - 2i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply each term: "},{"type":"inlineMath","attrs":{"latex":"3 \\times 8"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"3 \\times (-2i)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"-5i \\times 8"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"-5i \\times -2i"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify each product and combine like terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Replace "},{"type":"inlineMath","attrs":{"latex":"i^2"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":" where necessary."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1c. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\frac{5}{2 + 3i}"}},{"type":"text","marks":[{"type":"bold"}],"text":". Write your answer in the form a + bi."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Complex Numbers – Division and Rationalizing Denominators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to divide by a complex number and write the result in standard form by rationalizing the denominator."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To rationalize the denominator, multiply numerator and denominator by the conjugate of the denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conjugate of "},{"type":"inlineMath","attrs":{"latex":"a + bi"}},{"type":"text","text":" is "},{"type":"inlineMath","attrs":{"latex":"a - bi"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the conjugate of the denominator: "},{"type":"inlineMath","attrs":{"latex":"2 - 3i"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply numerator and denominator by the conjugate: "},{"type":"inlineMath","attrs":{"latex":"\\frac{5}{2 + 3i} \\times \\frac{2 - 3i}{2 - 3i}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the numerator: "},{"type":"inlineMath","attrs":{"latex":"5 \\times (2 - 3i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the denominator: "},{"type":"inlineMath","attrs":{"latex":"(2 + 3i)(2 - 3i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify both numerator and denominator, using "},{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1d. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\frac{1 + 6i}{5 - 7i}"}},{"type":"text","marks":[{"type":"bold"}],"text":". Write your answer in the form a + bi."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Complex Numbers – Division and Rationalizing Denominators"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to divide complex numbers and express the result in standard form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conjugate of "},{"type":"inlineMath","attrs":{"latex":"5 - 7i"}},{"type":"text","text":" is "},{"type":"inlineMath","attrs":{"latex":"5 + 7i"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply numerator and denominator by the conjugate to eliminate "},{"type":"inlineMath","attrs":{"latex":"i"}},{"type":"text","text":" from the denominator."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply numerator and denominator by the conjugate: "},{"type":"inlineMath","attrs":{"latex":"\\frac{1 + 6i}{5 - 7i} \\times \\frac{5 + 7i}{5 + 7i}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the numerator: "},{"type":"inlineMath","attrs":{"latex":"(1 + 6i)(5 + 7i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand the denominator: "},{"type":"inlineMath","attrs":{"latex":"(5 - 7i)(5 + 7i)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify both numerator and denominator, using "},{"type":"inlineMath","attrs":{"latex":"i^2 = -1"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2a. Solve the quadratic equation: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 8x + 15 = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations – Factoring"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve quadratic equations by factoring, the square root method, or the quadratic formula."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic equation: "},{"type":"inlineMath","attrs":{"latex":"ax^2 + bx + c = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring: Find two numbers that multiply to "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" and add to "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic formula: "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{ -b \\pm \\sqrt{b^2 - 4ac} }{2a}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"a = 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"b = -8"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"c = 15"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Look for two numbers that multiply to $15"},{"type":"inlineMath","attrs":{"latex":" and add to "}},{"type":"text","text":"-8$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the factored form: "},{"type":"inlineMath","attrs":{"latex":"(x - m)(x - n) = 0"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" are the numbers found."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set each factor equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2b. Solve the quadratic equation: "},{"type":"inlineMath","attrs":{"latex":"8x^2 - 22x = -5"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations – Quadratic Formula"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to rearrange and solve a quadratic equation using the quadratic formula."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic formula: "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{ -b \\pm \\sqrt{b^2 - 4ac} }{2a}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms to one side to set the equation to zero: "},{"type":"inlineMath","attrs":{"latex":"8x^2 - 22x + 5 = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"a = 8"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"b = -22"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"c = 5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug these values into the quadratic formula."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify under the square root: "},{"type":"inlineMath","attrs":{"latex":"b^2 - 4ac"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the two possible solutions using the "},{"type":"inlineMath","attrs":{"latex":"\\pm"}},{"type":"text","text":" symbol."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2c. Solve the quadratic equation: "},{"type":"inlineMath","attrs":{"latex":"4x^2 - 32 = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations – Square Root Method"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve a quadratic equation by isolating "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" and taking the square root."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Square root method: If "},{"type":"inlineMath","attrs":{"latex":"ax^2 = b"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"x = \\pm \\sqrt{\\frac{b}{a}}"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add $32"},{"type":"inlineMath","attrs":{"latex":" to both sides: $4x^2 = 32"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by $4"},{"type":"inlineMath","attrs":{"latex":" to isolate "}},{"type":"text","text":"x^2$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the square root of both sides, remembering to include "},{"type":"inlineMath","attrs":{"latex":"\\pm"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2d. 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After how many seconds is the ball at a height of 240 feet?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Quadratic Equations – Applications (Projectile Motion)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to set up and solve a quadratic equation in a real-world context."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Projectile motion formula: "},{"type":"inlineMath","attrs":{"latex":"S(t) = S_0 + v_0 t + \\frac{1}{2} a t^2"}},{"type":"text","text":" (here, "},{"type":"inlineMath","attrs":{"latex":"a = -32"}},{"type":"text","text":" ft/s"},{"type":"inlineMath","attrs":{"latex":"^2"}},{"type":"text","text":" for gravity, but the equation is already given)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"S(t) = 240"}},{"type":"text","text":" to find when the ball is at 240 feet: "},{"type":"inlineMath","attrs":{"latex":"192 + 64t - 16t^2 = 240"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $240$ from both sides to set the equation to zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rearrange terms to write the equation in standard quadratic form: "},{"type":"inlineMath","attrs":{"latex":"at^2 + bt + c = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" and set up the quadratic formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. A fourth-grade class decides to enclose a rectangular garden with a fence. The school is used as one of the sides of the rectangle. 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Solve: "},{"type":"inlineMath","attrs":{"latex":"x^4 - 8x^2 - 9 = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":". 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Solve: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{x - 1} = \\frac{3}{x + 2}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving rational expressions and to check for extraneous solutions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Cross-multiply to eliminate denominators."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for values that make denominators zero (extraneous solutions)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Cross-multiply: "},{"type":"inlineMath","attrs":{"latex":"2(x + 2) = 3(x - 1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand both sides and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if the solution makes any denominator zero."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7c. Solve: "},{"type":"inlineMath","attrs":{"latex":"\\frac{3y + 5}{y^2 + 5y} + \\frac{y + 4}{y + 5} = \\frac{y + 1}{y}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Rational Equations – Common Denominator"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve rational equations by finding a common denominator and checking for extraneous solutions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the least common denominator (LCD) for all terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides by the LCD to clear denominators."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for extraneous solutions by substituting back into the original equation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the LCD for "},{"type":"inlineMath","attrs":{"latex":"y^2 + 5y"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"y + 5"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply both sides by the LCD to eliminate denominators."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and solve the resulting equation for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for extraneous solutions by ensuring no denominator is zero for your solution(s)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7d. Solve: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{1 - 2x} = 3"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Radical Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving square roots and to check for extraneous solutions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{A} = B"}},{"type":"text","text":", square both sides: "},{"type":"inlineMath","attrs":{"latex":"A = B^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for extraneous solutions by substituting back into the original equation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Square both sides: "},{"type":"inlineMath","attrs":{"latex":"1 - 2x = 9"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check your solution in the original equation to ensure it is valid."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7e. Solve: "},{"type":"inlineMath","attrs":{"latex":"|4x - 3| + 1 = 7"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Absolute Value Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving absolute values."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To solve "},{"type":"inlineMath","attrs":{"latex":"|A| = B"}},{"type":"text","text":", set "},{"type":"inlineMath","attrs":{"latex":"A = B"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"A = -B"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract $1"},{"type":"inlineMath","attrs":{"latex":" from both sides: "}},{"type":"text","text":"|4x - 3| = 6$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up two equations: "},{"type":"inlineMath","attrs":{"latex":"4x - 3 = 6"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"4x - 3 = -6"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve each equation for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8a. Solve the absolute value inequality: "},{"type":"inlineMath","attrs":{"latex":"|3x + 4| < 13"}},{"type":"text","marks":[{"type":"bold"}],"text":". Show the final solution in interval notation and as a number line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Absolute Value Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"co