Q1. Simplify the following complex number expressions:

Background

Topic: Complex Numbers

This question tests your ability to perform arithmetic operations (addition, subtraction, multiplication) with complex numbers, and to simplify the results into standard form .

Key Terms and Formulas:

• Complex number: , where and are real numbers, and is the imaginary unit ().

• Standard form:

• Multiplication:

• Conjugate:

Step-by-Step Guidance

1. For each part, identify the real and imaginary components.

2. Apply the appropriate arithmetic operation (addition, subtraction, multiplication) to both real and imaginary parts.

3. Remember that when simplifying products involving .

4. Combine like terms to write the answer in standard form .

Try solving on your own before revealing the answer!

Q2. Solve the quadratic equation .

Background

Topic: Quadratic Equations

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

Key Terms and Formulas:

• Quadratic equation:

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

• Zero Product Property: If , then or .

Step-by-Step Guidance

1. Write the equation in standard form: .

2. Look for two numbers that multiply to and add to .

3. Factor the quadratic into two binomials.

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

Try solving on your own before revealing the answer!

Q3. Solve using the quadratic formula.

Background

Topic: Quadratic Formula

This question tests your ability to use the quadratic formula to solve equations that cannot be easily factored.

Key Terms and Formulas:

• Quadratic formula:

• Discriminant:

Step-by-Step Guidance

1. Rewrite the equation in standard form: .

2. Identify , , and from the equation.

3. Plug these values into the quadratic formula.

4. Calculate the discriminant and simplify under the square root.

Try solving on your own before revealing the answer!

Q4. Solve .

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor and solve quadratic equations where one term is missing.

Key Terms and Formulas:

• Factoring:

• Zero Product Property

Step-by-Step Guidance

1. Factor out the common term .

2. Set each factor equal to zero.

3. Solve for in each equation.

Try solving on your own before revealing the answer!

Q5. Solve .

Background

Topic: Difference of Squares

This question tests your ability to recognize and solve equations using the difference of squares formula.

Key Terms and Formulas:

• Difference of squares:

• Square root property

Step-by-Step Guidance

1. Rewrite the equation as .

2. Divide both sides by $9x^2$.

3. Take the square root of both sides, remembering to include both positive and negative roots.

Try solving on your own before revealing the answer!

Q6. Find the solutions to .

Background

Topic: Square Root Property

This question tests your ability to solve quadratic equations by taking square roots.

Key Terms and Formulas:

• Square root property:

Step-by-Step Guidance

1. Take the square root of both sides of the equation.

2. Express the answer as both positive and negative values.

Try solving on your own before revealing the answer!

Q7. Solve for .

Background

Topic: Quadratic Equations with Parameters

This question tests your ability to rearrange and solve quadratic equations that include parameters.

Key Terms and Formulas:

• Quadratic formula

• Factoring

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero.

2. Identify the quadratic form in .

3. Apply the quadratic formula or factoring as appropriate.

Try solving on your own before revealing the answer!

Q8. Solve for .

Background

Topic: Solving for a Variable

This question tests your ability to isolate a variable in an equation.

Key Terms and Formulas:

• Algebraic manipulation

Step-by-Step Guidance

1. Divide both sides by to isolate .

2. Add $4x$.

Try solving on your own before revealing the answer!

Q9. Solve .

Background

Topic: Higher Degree Equations

This question tests your ability to solve quartic equations by isolating the variable and taking roots.

Key Terms and Formulas:

• Quartic equation:

• Root extraction

Step-by-Step Guidance

1. Move $24.

2. Divide both sides by $8x^4$.

3. Take the fourth root of both sides.

Try solving on your own before revealing the answer!

Q10. Solve .

Background

Topic: Radical Equations

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

Key Terms and Formulas:

• Radical equation

• Squaring both sides

Step-by-Step Guidance

1. Isolate the square root term.

2. Square both sides to eliminate the radical.

3. Simplify and solve the resulting quadratic equation.

Try solving on your own before revealing the answer!