Q1. Solve quadratic equations by factoring.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by factoring, a fundamental algebraic technique.

Key Terms and Formulas:

• Quadratic equation:

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

Step-by-Step Guidance

1. Write the quadratic equation in standard form: .

2. Identify , , and .

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

4. Rewrite the middle term using these numbers and factor by grouping.

Try solving on your own before revealing the answer!

Q2. Solve quadratic equations by the square root property.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations using the square root property, which is useful when the equation is in the form .

Key Terms and Formulas:

• Square root property: If , then .

Step-by-Step Guidance

1. Isolate the squared term on one side of the equation.

2. Divide by the coefficient if necessary so you have .

3. Apply the square root property: .

Try solving on your own before revealing the answer!

Q3. Solve quadratic equations by completing the square.

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by completing the square, a method that rewrites the equation in a perfect square form.

Key Terms and Formulas:

• Completing the square: Transforming into .

• Perfect square trinomial: .

Step-by-Step Guidance

1. Move the constant term to the other side: .

2. If , divide both sides by .

3. Find and add it to both sides to complete the square.

4. Rewrite the left side as a squared binomial.

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Q4. Solve quadratic equations using the quadratic formula.

Background

Topic: Quadratic Equations

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

Key Terms and Formulas:

• Quadratic formula:

Step-by-Step Guidance

1. Write the equation in standard form: .

2. Identify , , and .

3. Plug these values into the quadratic formula.

4. Calculate the discriminant .

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Q5. Use the discriminant to determine the number and type of solutions to a given quadratic equation.

Background

Topic: Quadratic Equations

This question tests your ability to use the discriminant to analyze the nature of the solutions of a quadratic equation.

Key Terms and Formulas:

• Discriminant:

• If , two real solutions; if , one real solution; if , two complex solutions.

Step-by-Step Guidance

1. Write the equation in standard form and identify , , and .

2. Calculate the discriminant .

3. Analyze the sign of to determine the number and type of solutions.

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Q6. Solve radical equations.

Background

Topic: Radical Equations

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

Key Terms and Formulas:

• Radical equation: An equation with a variable inside a root.

• Isolate the radical, then raise both sides to the appropriate power.

Step-by-Step Guidance

1. Isolate the radical expression on one side of the equation.

2. Raise both sides to the power that eliminates the radical (e.g., square both sides for a square root).

3. Solve the resulting equation for the variable.

4. Check all solutions in the original equation to avoid extraneous solutions.

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Q7. Solve equations with rational exponents.

Background

Topic: Rational Exponents

This question tests your ability to solve equations where the variable is raised to a fractional power.

Key Terms and Formulas:

• Rational exponent: means the th root of raised to the th power.

• To solve, raise both sides to the reciprocal power.

Step-by-Step Guidance

1. Isolate the term with the rational exponent.

2. Raise both sides to the reciprocal of the exponent to solve for the variable.

3. Simplify and solve for the variable.

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Q8. Solve applications involving linear and absolute value inequalities.

Background

Topic: Linear and Absolute Value Inequalities

This question tests your ability to solve inequalities and interpret their solutions in applied contexts.

Key Terms and Formulas:

• Linear inequality:

• Absolute value inequality: or

Step-by-Step Guidance

1. For linear inequalities, isolate the variable and solve as you would a linear equation.

2. For absolute value inequalities, consider both cases: and .

3. Write the solution as an interval or set notation.

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Q9. Solve polynomial equations by factoring.

Background

Topic: Polynomial Equations

This question tests your ability to solve higher-degree polynomial equations by factoring.

Key Terms and Formulas:

• Polynomial equation:

• Factoring: Expressing as a product of factors.

Step-by-Step Guidance

1. Write the polynomial equation in standard form.

2. Factor the polynomial completely.

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

Try solving on your own before revealing the answer!

Q10. Solve equations involving absolute value.

Background

Topic: Absolute Value Equations

This question tests your ability to solve equations where the variable is inside an absolute value.

Key Terms and Formulas:

• Absolute value equation:

• Solution: or

Step-by-Step Guidance

1. Isolate the absolute value expression.

2. Set up two equations: one for and one for .

3. Solve each equation for .

Try solving on your own before revealing the answer!

Q11. Solve linear inequalities.

Background

Topic: Linear Inequalities

This question tests your ability to solve inequalities involving linear expressions.

Key Terms and Formulas:

• Linear inequality: or

Step-by-Step Guidance

1. Isolate the variable on one side of the inequality.

2. Solve for the variable as you would in a linear equation.

3. If you multiply or divide by a negative number, reverse the inequality sign.

Try solving on your own before revealing the answer!