Pre-Calculus Review for Calculus Preparation: Step-by-Step Guidance

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1a. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{49}"}}]},{"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: Square Roots"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to simplify square roots, especially perfect squares."}]},{"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 ("},{"type":"inlineMath","attrs":{"latex":"\\sqrt{x}"}},{"type":"text","text":"): The value that, when multiplied by itself, gives "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Perfect square: A number that is the square of an integer."}]}]}]},{"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 if 49 is a perfect square. Ask yourself: Is there an integer "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"n^2 = 49"}},{"type":"text","text":"?"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the definition of a square root: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{49}"}},{"type":"text","text":" is the number that, when squared, equals 49."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Think about what integer squared gives 49. Set up "},{"type":"inlineMath","attrs":{"latex":"n^2 = 49"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"n"}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{49} = 7"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"49 is a perfect square because "},{"type":"inlineMath","attrs":{"latex":"7^2 = 49"}},{"type":"text","text":", so the square root is 7."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1b. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{50}"}}]},{"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: Simplifying Square Roots"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify square roots that are not perfect squares by factoring."}]},{"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 ("},{"type":"inlineMath","attrs":{"latex":"\\sqrt{x}"}},{"type":"text","text":")"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factorization: Expressing a number as a product of its factors."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplifying: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{ab} = \\sqrt{a} \\times \\sqrt{b}"}}]}]}]},{"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":"Factor 50 into its prime factors: "},{"type":"inlineMath","attrs":{"latex":"50 = 2 \\times 25"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recognize that 25 is a perfect square: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{25} = 5"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the property "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{ab} = \\sqrt{a} \\times \\sqrt{b}"}},{"type":"text","text":" to rewrite "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{50}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the expression for simplification, but stop before calculating the final value."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{50} = 5\\sqrt{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"We factored 50 as "},{"type":"inlineMath","attrs":{"latex":"2 \\times 25"}},{"type":"text","text":", so "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{50} = \\sqrt{25} \\times \\sqrt{2} = 5\\sqrt{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1c. Simplify: "},{"type":"inlineMath","attrs":{"latex":"7 \\pm \\sqrt{100}"}}]},{"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: Square Roots and Arithmetic"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate square roots and perform addition/subtraction."}]},{"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":"inlineMath","attrs":{"latex":"\\pm"}},{"type":"text","text":" means \"plus or minus\"; you need to consider both "},{"type":"inlineMath","attrs":{"latex":"+"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-"}},{"type":"text","text":" cases."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{100}"}},{"type":"text","text":" is a perfect square."}]}]}]},{"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":"Calculate "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{100}"}},{"type":"text","text":" by identifying the integer whose square is 100."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute the value of "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{100}"}},{"type":"text","text":" into the expression "},{"type":"inlineMath","attrs":{"latex":"7 \\pm \\sqrt{100}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write out both cases: "},{"type":"inlineMath","attrs":{"latex":"7 + \\sqrt{100}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"7 - \\sqrt{100}"}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"7 \\pm \\sqrt{100} = 7 \\pm 10"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This gives two values: "},{"type":"inlineMath","attrs":{"latex":"7 + 10 = 17"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"7 - 10 = -3"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1d. Simplify: "},{"type":"inlineMath","attrs":{"latex":"\\frac{4 \\pm \\sqrt{28}}{4}"}}]},{"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: Simplifying Radical Expressions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify square roots and rational expressions."}]},{"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":"inlineMath","attrs":{"latex":"\\sqrt{28}"}},{"type":"text","text":" can be simplified by factoring."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fraction: "},{"type":"inlineMath","attrs":{"latex":"\\frac{a \\pm b}{c}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplifying radicals: "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{ab} = \\sqrt{a} \\times \\sqrt{b}"}}]}]}]},{"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":"Factor 28 into "},{"type":"inlineMath","attrs":{"latex":"4 \\times 7"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{28}"}},{"type":"text","text":" using "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{4} = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{28}"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"2\\sqrt{7}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"2\\sqrt{7}"}},{"type":"text","text":" into the original expression and simplify the fraction, but stop before the final step."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{4 \\pm \\sqrt{28}}{4} = 1 \\pm \\frac{\\sqrt{7}}{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"After simplifying "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{28}"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"2\\sqrt{7}"}},{"type":"text","text":", the expression becomes "},{"type":"inlineMath","attrs":{"latex":"\\frac{4 \\pm 2\\sqrt{7}}{4} = 1 \\pm \\frac{\\sqrt{7}}{2}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2a. Distribute and simplify: "},{"type":"inlineMath","attrs":{"latex":"-5x^2(3x-5)"}}]},{"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: Distributive Property"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to distribute a monomial across a binomial and combine like terms."}]},{"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":"Distributive property: "},{"type":"inlineMath","attrs":{"latex":"a(b + c) = ab + ac"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Monomial: A single term (e.g., "},{"type":"inlineMath","attrs":{"latex":"-5x^2"}},{"type":"text","text":")"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Binomial: Two terms (e.g., "},{"type":"inlineMath","attrs":{"latex":"3x - 5"}},{"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: Multiply "},{"type":"inlineMath","attrs":{"latex":"-5x^2"}},{"type":"text","text":" by each term inside the parentheses."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"-5x^2 \\times 3x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"-5x^2 \\times (-5)"}},{"type":"text","text":" separately."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the resulting terms and combine them into a single expression."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"-5x^2(3x-5) = -15x^3 + 25x^2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each term was multiplied and combined correctly."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2b. Expand and simplify: "},{"type":"inlineMath","attrs":{"latex":"(3x-4)(x-2)"}}]},{"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: FOIL Method (First, Outer, Inner, Last)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to multiply two binomials and combine like terms."}]},{"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: Multiply First, Outer, Inner, Last terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Binomial multiplication."}]}]}]},{"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 the first terms: "},{"type":"inlineMath","attrs":{"latex":"3x \\times x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the outer terms: "},{"type":"inlineMath","attrs":{"latex":"3x \\times (-2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the inner terms: "},{"type":"inlineMath","attrs":{"latex":"-4 \\times x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the last terms: "},{"type":"inlineMath","attrs":{"latex":"-4 \\times (-2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combine all terms and simplify, but stop before the final simplification."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"(3x-4)(x-2) = 3x^2 - 10x + 8"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All terms were multiplied and combined correctly."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2c. Expand and simplify: "},{"type":"inlineMath","attrs":{"latex":"(2x^3 + 7)^2"}}]},{"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: Binomial Squaring"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to expand the square of a binomial using the formula "},{"type":"inlineMath","attrs":{"latex":"(a+b)^2 = a^2 + 2ab + b^2"}},{"type":"text","text":"."}]},{"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":"Binomial square: "},{"type":"inlineMath","attrs":{"latex":"(a+b)^2 = a^2 + 2ab + b^2"}}]}]}]},{"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 = 2x^3"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b = 7"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the binomial square formula: "},{"type":"inlineMath","attrs":{"latex":"(a+b)^2 = a^2 + 2ab + b^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"a^2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"2ab"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"b^2"}},{"type":"text","text":" separately."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the expanded expression, but stop before combining like terms."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"(2x^3 + 7)^2 = 4x^6 + 28x^3 + 49"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each term was calculated using the binomial square formula."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3a. Factor: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 16"}}]},{"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: Factoring Difference of Squares"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to recognize and factor expressions of the form "},{"type":"inlineMath","attrs":{"latex":"a^2 - b^2"}},{"type":"text","text":"."}]},{"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":"Difference of squares: "},{"type":"inlineMath","attrs":{"latex":"a^2 - b^2 = (a-b)(a+b)"}}]}]}]},{"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":"x^2"}},{"type":"text","text":" and $16$ as perfect squares."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the difference of squares formula: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 16 = (x - 4)(x + 4)"}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 16 = (x - 4)(x + 4)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factored using the difference of squares formula."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3b. Factor: "},{"type":"inlineMath","attrs":{"latex":"25x^2 - 49"}}]},{"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: Factoring Difference of Squares"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to factor expressions where both terms are perfect squares."}]},{"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":"Difference of squares: "},{"type":"inlineMath","attrs":{"latex":"a^2 - b^2 = (a-b)(a+b)"}}]}]}]},{"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":"25x^2"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"(5x)^2"}},{"type":"text","text":" and $49"},{"type":"inlineMath","attrs":{"latex":" as $7^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the difference of squares formula: "},{"type":"inlineMath","attrs":{"latex":"25x^2 - 49 = (5x - 7)(5x + 7)"}},{"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":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"25x^2 - 49 = (5x - 7)(5x + 7)"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Both terms are perfect squares, so the formula applies directly."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3c. 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