Q1. Find the sum: (2 - i) - (-3 + 5i)

Background

Topic: Complex Numbers – Addition and Subtraction

This question tests your ability to add and subtract complex numbers, combining real and imaginary parts.

Key Terms and Formulas:

• Complex number: where is the real part and is the imaginary part.

• Add/subtract: Combine real parts and imaginary parts separately.

Step-by-Step Guidance

1. Write both complex numbers: and .

2. Subtract the second from the first: .

3. Distribute the negative sign: .

4. Combine real parts: ; combine imaginary parts: .

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Q2. Multiply: (-4 - 3i)(5 - 6i)

Background

Topic: Complex Numbers – Multiplication

This question tests your ability to multiply two complex numbers using the distributive property (FOIL method).

Key Terms and Formulas:

• Complex number:

• Multiplication:

Step-by-Step Guidance

1. Write both numbers: and .

2. Apply FOIL: Multiply each term in the first by each in the second.

3. Calculate: , , , .

4. Combine like terms and use to simplify.

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Q3. Find the product of and its conjugate

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Topic: Complex Numbers – Conjugates

This question tests your understanding of multiplying a complex number by its conjugate.

Key Terms and Formulas:

• Conjugate: If , then conjugate is .

• Product:

Step-by-Step Guidance

1. Write the complex number: .

2. Write its conjugate: .

3. Multiply: .

4. Apply the formula: where , .

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Q4. Divide:

Background

Topic: Complex Numbers – Division

This question tests your ability to divide complex numbers by rationalizing the denominator.

Key Terms and Formulas:

• To divide by , multiply numerator and denominator by the conjugate .

• Conjugate:

• Formula:

Step-by-Step Guidance

1. Write the expression: .

2. Multiply numerator and denominator by the conjugate of the denominator: .

3. Calculate numerator: .

4. Calculate denominator: .

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Q5. Write in the form

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Topic: Complex Numbers – Square Roots of Negative Numbers

This question tests your ability to simplify square roots of negative numbers and write the result as a complex number.

Key Terms and Formulas:

• Combine real and imaginary parts.

Step-by-Step Guidance

1. Simplify .

2. Simplify using .

3. Combine the results: real part and imaginary part.

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Q6. Write in the form

Background

Topic: Complex Numbers – Square Roots of Negative Numbers

This question tests your ability to simplify and write expressions involving square roots of negative numbers as complex numbers.

Key Terms and Formulas:

• Combine real and imaginary parts.

Step-by-Step Guidance

1. Simplify (real part).

2. Simplify : , so .

3. Combine: .

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Q7. Solve by factoring:

Background

Topic: Quadratic Equations – Factoring

This question tests your ability to solve quadratic equations by factoring.

Key Terms and Formulas:

• Quadratic equation:

• Factoring: Set each factor equal to zero.

Step-by-Step Guidance

1. Write the equation: .

2. Factor out : .

3. Set each factor equal to zero: or .

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Q8. Solve by factoring:

Background

Topic: Quadratic Equations – Factoring

This question tests your ability to factor and solve quadratic equations.

Key Terms and Formulas:

• Quadratic equation:

• Factoring: Find two numbers that multiply to and add to .

Step-by-Step Guidance

1. Write the equation: .

2. Factor: Find two numbers that multiply to $15.

3. Write the factors: .

4. Set each factor equal to zero and solve for .

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Q9. Solve by factoring:

Background

Topic: Quadratic Equations – Factoring

This question tests your ability to rearrange and factor quadratic equations.

Key Terms and Formulas:

• Quadratic equation:

• Factoring: Rearranging and factoring.

Step-by-Step Guidance

1. Move all terms to one side: .

2. Factor the quadratic equation.

3. Set each factor equal to zero and solve for .

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Q10. Solve using the square root property:

Background

Topic: Quadratic Equations – Square Root Property

This question tests your ability to solve quadratic equations using the square root property.

Key Terms and Formulas:

• Square root property: If , then .

Step-by-Step Guidance

1. Divide both sides by $4(x - 2)^2$.

2. Take the square root of both sides: .

3. Solve for by adding $2$ to both sides.

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Q11. Solve by completing the square:

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Topic: Quadratic Equations – Completing the Square

This question tests your ability to solve quadratic equations by completing the square.

Key Terms and Formulas:

• Completing the square:

• Move constant to other side, add to both sides.

Step-by-Step Guidance

1. Move constant to other side: .

2. Add to both sides.

3. Write as a perfect square: .

4. Take the square root: .

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Q12. Solve using the quadratic formula:

Background

Topic: Quadratic Equations – Quadratic Formula

This question tests your ability to use the quadratic formula to solve equations.

Key Terms and Formulas:

• Quadratic formula:

• Discriminant:

Step-by-Step Guidance

1. Identify , , .

2. Calculate discriminant: .

3. Plug values into quadratic formula.

4. Simplify under the square root and denominator.

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Q13. Use the discriminant to determine the number and nature of solutions for

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Topic: Quadratic Equations – Discriminant

This question tests your ability to use the discriminant to determine the nature of solutions.

Key Terms and Formulas:

• Discriminant:

• If discriminant , two real solutions; , one real solution; , two nonreal solutions.

Step-by-Step Guidance

1. Identify , , .

2. Calculate discriminant: .

3. Determine if discriminant is positive, zero, or negative.

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Q14. The sum of the square of a positive number and the square of 3 more than the number is 149. What is the number?

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Topic: Quadratic Equations – Word Problems

This question tests your ability to translate a word problem into a quadratic equation and solve it.

Key Terms and Formulas:

• Let be the positive number.

• Equation:

Step-by-Step Guidance

1. Let be the positive number.

2. Write the equation: .

3. Expand .

4. Combine like terms and solve the quadratic equation.

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Q15. Benjamin threw a rock from a cliff 36 ft above water. Height after seconds: . How long until it hits the water?

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Topic: Quadratic Equations – Applications

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

Key Terms and Formulas:

• Set to find when the rock hits the water.

• Equation:

Step-by-Step Guidance

1. Set and write the equation: .

2. Solve the quadratic equation for using factoring or quadratic formula.

3. Choose the positive value for (time cannot be negative).

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Q16. A kite is flying on 106 ft of string. Its height is 34 ft more than the horizontal distance from the person. How high is it?

Background

Topic: Quadratic Equations – Applications

This question tests your ability to set up and solve a right triangle problem using the Pythagorean theorem.

Key Terms and Formulas:

• Let be the horizontal distance.

• Height:

• Pythagorean theorem:

Step-by-Step Guidance

1. Let be the horizontal distance; height is .

2. Write the equation: .

3. Expand and solve for .

4. Find the height: .

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Q17. Find all solutions:

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Topic: Polynomial Equations – Cubic Equations

This question tests your ability to solve cubic equations by factoring.

Key Terms and Formulas:

• Move all terms to one side:

• Factor the cubic equation.

Step-by-Step Guidance

1. Move all terms to one side: .

2. Factor the cubic equation (try rational root theorem or grouping).

3. Set each factor equal to zero and solve for .

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Q18. Solve the radical equation:

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Topic: Radical Equations

This question tests your ability to solve equations involving square roots.

Key Terms and Formulas:

• Set the radicand equal to zero:

Step-by-Step Guidance

1. Set .

2. Square both sides to eliminate the radical.

3. Solve for .

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Q19. Solve the polynomial inequality:

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Topic: Polynomial Inequalities

This question tests your ability to solve polynomial inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Move all terms to one side:

• Factor and solve the inequality.

Step-by-Step Guidance

1. Move all terms to one side: .

2. Factor the expression.

3. Find critical points and test intervals.

4. Express the solution in interval notation.

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Q20. Solve the rational inequality:

Background

Topic: Rational Inequalities

This question tests your ability to solve rational inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Find zeros of numerator and denominator.

• Test intervals between critical points.

• Express solution in interval notation.

Step-by-Step Guidance

1. Set numerator and denominator to find critical points.

2. Test values in each interval to see where the inequality holds.

3. Include endpoints as appropriate.

Try solving on your own before revealing the answer!