Q1. Solve for x:

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values, which may yield multiple solutions, including complex numbers.

Key Terms and Formulas:

• Absolute value: means or

Step-by-Step Guidance

1. Start by isolating the absolute value:

2. Set up two equations: and

3. Solve each quadratic equation separately for .

4. For each equation, use the quadratic formula:

Try solving on your own before revealing the answer!

Q2. Solve for x:

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations involving absolute values and recognize when no real solution exists.

Key Terms and Formulas:

• Absolute value: for any real

Step-by-Step Guidance

1. Isolate the absolute value:

2. Consider whether can be negative for real .

3. If not, consider the possibility of complex solutions.

Try solving on your own before revealing the answer!

Q3. Solve the inequality:

Background

Topic: Absolute Value Inequalities

This question tests your understanding of inequalities involving absolute values and their solution sets.

Key Terms and Formulas:

• Absolute value: for any real

Step-by-Step Guidance

1. Isolate the absolute value:

2. Consider whether the inequality can be satisfied for any real .

3. Think about the graph of and where it is less than a negative number.

Try solving on your own before revealing the answer!

Q4. Solve the inequality:

Background

Topic: Quadratic Inequalities

This question tests your ability to solve quadratic inequalities and represent the solution on the real number line.

Key Terms and Formulas:

• Quadratic expression:

• Factoring or quadratic formula to find critical points

Step-by-Step Guidance

1. Rewrite the inequality:

2. Find the roots of using factoring or the quadratic formula.

3. Determine the intervals where the quadratic is positive.

4. Sketch the solution on the real number line.

Try solving on your own before revealing the answer!

Q5. Solve for x:

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations, including those with complex solutions.

Key Terms and Formulas:

• Quadratic formula:

Step-by-Step Guidance

1. Move all terms to one side:

2. Simplify and write in standard form:

3. Identify , , and for the quadratic formula.

4. Set up the quadratic formula for .

Try solving on your own before revealing the answer!

Q6. Solve for x:

Background

Topic: Quadratic Equations

This question tests your ability to expand, simplify, and solve quadratic equations.

Key Terms and Formulas:

• Quadratic formula:

Step-by-Step Guidance

1. Expand and add 7.

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

3. Simplify to standard quadratic form.

4. Identify , , and for the quadratic formula.

Try solving on your own before revealing the answer!

Q7. Compute given and

Background

Topic: Complex Number Arithmetic

This question tests your ability to multiply complex numbers using algebraic rules.

Key Terms and Formulas:

• Complex multiplication:

Step-by-Step Guidance

1. Write and in standard form.

2. Multiply using distributive property.

3. Combine like terms and simplify.

Try solving on your own before revealing the answer!

Q8. Compute given and

Background

Topic: Complex Number Arithmetic

This question tests your ability to perform operations with complex numbers, including subtraction, multiplication, and addition.

Key Terms and Formulas:

• Complex addition and subtraction:

• Complex multiplication:

Step-by-Step Guidance

1. Compute .

2. Multiply the result by .

3. Add to the result.

Try solving on your own before revealing the answer!