Q1. What is a quadratic equation? Give your own definition and examples.

Background

Topic: Quadratic Equations

This question is testing your understanding of what constitutes a quadratic equation and your ability to recognize and create examples.

Key Terms:

• Quadratic Equation: An equation of the form , where .

Step-by-Step Guidance

1. Think about the general form of a quadratic equation: .

2. Identify what makes an equation quadratic (the highest power of is 2).

3. Try to write your own definition using these ideas.

4. Come up with at least two examples that fit this form.

Try solving on your own before revealing the answer!

Q2. Solve by factoring: (a) (b)

Background

Topic: Solving Quadratic Equations by Factoring

This question is testing your ability to solve quadratic equations by rearranging and factoring.

Key Terms and Formulas:

• Zero-Product Principle: If , then or .

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

Step-by-Step Guidance

1. Rewrite each equation in standard form ().

2. Factor the quadratic expression.

3. Apply the zero-product principle to set each factor equal to zero.

4. Solve for in each resulting equation.

Try solving on your own before revealing the answer!

Q3. Solve by factoring: (a) (b)

Background

Topic: Solving Quadratic Equations by Factoring

This question is testing your ability to factor and solve quadratic equations.

Key Terms and Formulas:

• Standard Form:

• Factoring: Breaking down the quadratic into products.

Step-by-Step Guidance

1. Rewrite each equation in standard form.

2. Factor the quadratic expression.

3. Set each factor equal to zero.

4. Solve for in each equation.

Try solving on your own before revealing the answer!

Q4. Solve by the square root method: (a) (b) (c)

Background

Topic: Solving Quadratic Equations by the Square Root Property

This question is testing your ability to isolate the squared term and use the square root property.

Key Terms and Formulas:

• Square Root Property: If , then .

Step-by-Step Guidance

1. Isolate the squared term in each equation.

2. Divide or subtract as needed to get or alone.

3. Apply the square root property to solve for .

4. Remember to consider both the positive and negative roots.

Try solving on your own before revealing the answer!

Q5. Solve by the square root method: (a) (b) (c)

Background

Topic: Solving Quadratic Equations by the Square Root Property

This question is testing your ability to manipulate and solve quadratics using the square root method.

Key Terms and Formulas:

• Square Root Property:

Step-by-Step Guidance

1. Isolate the squared term in each equation.

2. Divide or subtract as needed to get or alone.

3. Apply the square root property to solve for .

4. Remember to consider both positive and negative roots.

Try solving on your own before revealing the answer!

Q6. What term should be added to the binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

Background

Topic: Completing the Square

This question is testing your ability to create a perfect square trinomial by adding the correct term.

Key Terms and Formulas:

• Perfect Square Trinomial:

Step-by-Step Guidance

1. Identify the coefficient in .

2. Calculate .

3. Add this value to the binomial to form a perfect square trinomial.

4. Factor the resulting trinomial.

Try solving on your own before revealing the answer!

Q7. Solve by completing the square: (a) (b)

Background

Topic: Solving Quadratic Equations by Completing the Square

This question is testing your ability to solve quadratics by completing the square.

Key Terms and Formulas:

• Completing the Square: Transforming into

Step-by-Step Guidance

1. Rewrite the equation so the and terms are on one side.

2. If the coefficient of is not 1, divide the whole equation by that coefficient.

3. Find the value to complete the square: .

4. Add and subtract this value as needed, then rewrite as a squared binomial.

Try solving on your own before revealing the answer!

Q8. Solve by completing the square: (a) (b)

Background

Topic: Solving Quadratic Equations by Completing the Square

This question is testing your ability to use the completing the square method.

Key Terms and Formulas:

• Completing the Square:

Step-by-Step Guidance

1. Rewrite the equation in standard form.

2. If needed, divide by the coefficient of .

3. Find the value to complete the square: .

4. Add and subtract this value, then rewrite as a squared binomial.

Try solving on your own before revealing the answer!

Q9. Solve using the quadratic formula: (a) (b)

Background

Topic: Quadratic Formula

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

Key Terms and Formulas:

• Quadratic Formula:

Step-by-Step Guidance

1. Identify , , and from the equation.

2. Plug these values into the quadratic formula.

3. Calculate the discriminant .

4. Set up the formula for the final calculation.

Try solving on your own before revealing the answer!

Q10. Solve using the quadratic formula: (a) (b)

Background

Topic: Quadratic Formula

This question is testing your ability to solve quadratic equations using the quadratic formula.

Key Terms and Formulas:

• Quadratic Formula:

Step-by-Step Guidance

1. Identify , , and for each equation.

2. Plug these values into the quadratic formula.

3. Calculate the discriminant .

4. Set up the formula for the final calculation.

Try solving on your own before revealing the answer!

Q11. Compute the discriminant and determine the number and types of solutions: (a) (b) (c)

Background

Topic: Discriminant of Quadratic Equations

This question is testing your ability to compute the discriminant and interpret the nature of the solutions.

Key Terms and Formulas:

• Discriminant:

• Types of Solutions: If , two real solutions; , one real solution; , two complex solutions.

Step-by-Step Guidance

1. Identify , , and for each equation.

2. Calculate the discriminant .

3. Interpret the value of to determine the number and type of solutions.

Try solving on your own before revealing the answer!

Q12. Compute the discriminant and determine the number and types of solutions: (a) (b) (c)

Background

Topic: Discriminant of Quadratic Equations

This question is testing your ability to use the discriminant to classify solutions.

Key Terms and Formulas:

• Discriminant:

Step-by-Step Guidance

1. Identify , , and for each equation.

2. Calculate the discriminant .

3. Interpret the value of to determine the number and type of solutions.

Try solving on your own before revealing the answer!

Q13. The function models a woman’s normal systolic blood pressure, , at age . Use this function to find the age, to the nearest year, of a woman whose normal systolic blood pressure is 115 mmHg.

Background

Topic: Quadratic Application Problems

This question is testing your ability to set up and solve a quadratic equation in a real-world context.

Key Terms and Formulas:

• Quadratic Equation:

Step-by-Step Guidance

1. Set the function equal to 115: .

2. Subtract 115 from both sides to get 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!

Q14. The number of fatal crashes per 100 million miles driven, , for drivers of age can be modeled by . What age groups are expected to be in 3 fatal crashes per 100 million miles driven?

Background

Topic: Quadratic Application Problems

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

Key Terms and Formulas:

• Quadratic Equation:

Step-by-Step Guidance

1. Set the function equal to 3: .

2. Subtract 3 from both sides to get 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!