College Algebra Test #1 Review – Step-by-Step Study Guidance

Q1. Write the quotient in the standard form a + bi: \( \frac{5}{-9i} \)

Background

Topic: Complex Numbers – Division

This question tests your ability to write a complex quotient in standard form \( a + bi \) by rationalizing the denominator.

Key Terms and Formulas:

• Standard form of a complex number: \( a + bi \)

• To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator (if necessary).

• For pure imaginary denominators, multiply numerator and denominator by \( i \) or \( -i \) to rationalize.

Step-by-Step Guidance

1. Recognize that the denominator is \( -9i \), a pure imaginary number.

2. To eliminate the imaginary unit from the denominator, multiply both numerator and denominator by \( i \).

3. Set up the multiplication: \( \frac{5}{-9i} \times \frac{i}{i} \).

4. Multiply out the numerator and denominator, remembering that \( i^2 = -1 \).

Try solving on your own before revealing the answer!

Final Answer: \( -\frac{5}{9}i \)

Multiplying numerator and denominator by \( i \) gives \( \frac{5i}{-9i^2} = \frac{5i}{9} \), and since \( i^2 = -1 \), the denominator becomes 9. The standard form is \( 0 - \frac{5}{9}i \) or simply \( -\frac{5}{9}i \).

Q2. Write the quotient in the standard form a + bi: \( \frac{5i}{2 + i} \)

Background

Topic: Complex Numbers – Division

This question tests your ability to divide complex numbers and express the result in standard form \( a + bi \).

Key Terms and Formulas:

• Standard form: \( a + bi \)

• Conjugate: For \( 2 + i \), the conjugate is \( 2 - i \).

• Multiply numerator and denominator by the conjugate of the denominator to rationalize.

Step-by-Step Guidance

1. Identify the denominator \( 2 + i \) and its conjugate \( 2 - i \).

2. Multiply numerator and denominator by \( 2 - i \): \( \frac{5i}{2 + i} \times \frac{2 - i}{2 - i} \).

3. Expand the numerator: \( 5i(2 - i) \).

4. Expand the denominator: \( (2 + i)(2 - i) \).

5. Simplify both numerator and denominator, using \( i^2 = -1 \).

Try solving on your own before revealing the answer!

Final Answer: \( 1 + 2i \)

After multiplying and simplifying, the result is \( 1 + 2i \) in standard form.

Q3. Solve the equation by factoring: \( 8x^2 - 71x = 9 \)

Background

Topic: Quadratic Equations – Factoring

This question tests your ability to solve a quadratic equation by moving all terms to one side and factoring.

Key Terms and Formulas:

• Quadratic equation: \( ax^2 + bx + c = 0 \)

• Factoring: Expressing the quadratic as a product of two binomials.

• Zero Product Property: If \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).

Step-by-Step Guidance

1. Move all terms to one side: \( 8x^2 - 71x - 9 = 0 \).

2. Look for two numbers that multiply to \( 8 \times (-9) = -72 \) and add to \( -71 \).

3. Factor the quadratic using these numbers, or use factoring by grouping if necessary.

4. Set each factor equal to zero to solve for \( x \).

Try solving on your own before revealing the answer!

Final Answer: \( x = -\frac{1}{8},\ x = 9 \)

Factoring and solving gives two real solutions: \( x = -\frac{1}{8} \) and \( x = 9 \).

Q4. Solve the equation by completing the square: \( 4x^2 - 3x + 1 = 0 \)

Background

Topic: Quadratic Equations – Completing the Square

This question tests your ability to solve a quadratic equation by rewriting it in a perfect square form.

Key Terms and Formulas:

• Completing the square: Transforming \( ax^2 + bx + c = 0 \) into \( (x + d)^2 = e \).

• Square root property: If \( x^2 = k \), then \( x = \pm \sqrt{k} \).

Step-by-Step Guidance

1. Move the constant to the other side: \( 4x^2 - 3x = -1 \).

2. Divide both sides by 4 to make the coefficient of \( x^2 \) equal to 1.

3. Identify the value to complete the square: take half of the coefficient of \( x \), square it, and add to both sides.

4. Rewrite the left side as a perfect square trinomial.

5. Take the square root of both sides and solve for \( x \).

Try solving on your own before revealing the answer!

Final Answer: \( x = \frac{3}{8} \pm \frac{i\sqrt{7}}{8} \)

Completing the square and solving gives two complex solutions: \( x = \frac{3}{8} + \frac{i\sqrt{7}}{8} \) and \( x = \frac{3}{8} - \frac{i\sqrt{7}}{8} \).

Q5. Solve the equation using the quadratic formula: \( (2x - 1)(x + 1) = 6 \)

Background

Topic: Quadratic Equations – Quadratic Formula

This question tests your ability to expand, rearrange, and solve a quadratic equation using the quadratic formula.

Key Terms and Formulas:

• Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

• Standard form: \( ax^2 + bx + c = 0 \)

Step-by-Step Guidance

1. Expand the left side: \( (2x - 1)(x + 1) = 2x^2 + 2x - x - 1 = 2x^2 + x - 1 \).

2. Set the equation to zero: \( 2x^2 + x - 1 - 6 = 0 \Rightarrow 2x^2 + x - 7 = 0 \).

3. Identify \( a = 2, b = 1, c = -7 \).

4. Plug these values into the quadratic formula.

5. Simplify under the square root (the discriminant) and set up the solution for \( x \).

Try solving on your own before revealing the answer!

Final Answer: \( x = \frac{-1 + \sqrt{57}}{4},\ x = \frac{-1 - \sqrt{57}}{4} \)

Using the quadratic formula with the identified coefficients gives two real solutions.

Q6. Find the discriminant and determine the number and type of roots: \( x^2 - 4x + 3 = 0 \)

Background

Topic: Quadratic Equations – Discriminant

This question tests your ability to compute the discriminant and interpret the nature of the roots of a quadratic equation.

Key Terms and Formulas:

• Discriminant: \( D = b^2 - 4ac \)

• If \( D > 0 \): two real, unequal roots; \( D = 0 \): one real, repeated root; \( D < 0 \): two complex roots.

Step-by-Step Guidance

1. Identify \( a = 1, b = -4, c = 3 \).

2. Calculate the discriminant: \( D = (-4)^2 - 4(1)(3) \).

3. Interpret the value of \( D \) to determine the number and type of roots.

Try solving on your own before revealing the answer!

Final Answer: \( D = 4 \), two real, unequal roots

The discriminant is positive, so the equation has two real, unequal roots.

Q7. Find the discriminant and determine the number and type of roots: \( 8y^2 = -3y - 7 \)

Background

Topic: Quadratic Equations – Discriminant

This question tests your ability to rewrite a quadratic equation in standard form, compute the discriminant, and interpret the roots.

Key Terms and Formulas:

• Standard form: \( ay^2 + by + c = 0 \)

• Discriminant: \( D = b^2 - 4ac \)

Step-by-Step Guidance

1. Move all terms to one side: \( 8y^2 + 3y + 7 = 0 \).

2. Identify \( a = 8, b = 3, c = 7 \).

3. Calculate the discriminant: \( D = 3^2 - 4 \times 8 \times 7 \).

4. Interpret the value of \( D \) to determine the number and type of roots.

Try solving on your own before revealing the answer!

Final Answer: \( D = -215 \), two complex roots

The discriminant is negative, so the equation has two complex roots.

Q8. Find the discriminant and determine the number and type of roots: \( t^2 - 18t + 81 = 0 \)

Background

Topic: Quadratic Equations – Discriminant

This question tests your ability to compute the discriminant and interpret the roots for a quadratic equation.

Key Terms and Formulas:

• Discriminant: \( D = b^2 - 4ac \)

Step-by-Step Guidance

1. Identify \( a = 1, b = -18, c = 81 \).

2. Calculate the discriminant: \( D = (-18)^2 - 4 \times 1 \times 81 \).

3. Interpret the value of \( D \) to determine the number and type of roots.

Try solving on your own before revealing the answer!

Final Answer: \( D = 0 \), one real, repeated root

The discriminant is zero, so the equation has one real, repeated root.

Q9. The length of a rectangular storage room is 2 feet longer than its width. If the area of the room is 120 square feet, find its dimensions.

Background

Topic: Quadratic Equations – Applications (Geometry)

This question tests your ability to set up and solve a quadratic equation from a geometric word problem.

Key Terms and Formulas:

• Area of rectangle: \( \text{Area} = \text{length} \times \text{width} \)

• Let width = \( x \), length = \( x + 2 \)

Step-by-Step Guidance

1. Let the width be \( x \) feet, so the length is \( x + 2 \) feet.

2. Set up the equation: \( x(x + 2) = 120 \).

3. Expand and rearrange: \( x^2 + 2x - 120 = 0 \).

4. Solve the quadratic equation for \( x \) (width), then find the length.

Try solving on your own before revealing the answer!

Final Answer: 10 ft by 12 ft

Solving the quadratic gives width = 10 ft, length = 12 ft.

Q10. A toy rocket is shot vertically upward from the ground. Its distance in feet from the ground in t seconds is given by \( s(t) = -16t^2 + 171t \). At what time(s) will the rocket be 118 ft from the ground? Round your answer to the nearest tenth, if necessary.

Background

Topic: Quadratic Equations – Applications (Projectile Motion)

This question tests your ability to solve a quadratic equation in a real-world context and interpret the solutions.

Key Terms and Formulas:

• Projectile motion equation: \( s(t) = -16t^2 + v_0 t + s_0 \)

• Set \( s(t) = 118 \) and solve for \( t \).

Step-by-Step Guidance

1. Set up the equation: \( -16t^2 + 171t = 118 \).

2. Move all terms to one side: \( -16t^2 + 171t - 118 = 0 \).

3. Identify coefficients: \( a = -16, b = 171, c = -118 \).

4. Plug into the quadratic formula and simplify under the square root.

5. Solve for \( t \), rounding to the nearest tenth as needed.

Try solving on your own before revealing the answer!

Final Answer: 0.7 sec and 9.9 sec

The rocket is 118 ft from the ground at approximately 0.7 seconds and 9.9 seconds after launch.