Q: A piece of property is a right triangle. The longer leg is 20 m longer than twice the shorter leg. The hypotenuse is 10 m longer than the longer leg. If the shorter leg is $$x$$, what is the equation for the hypotenuse?

$$\text{Hypotenuse} = 2x + 30$$

Question 1

Which of the following is the general form of a quadratic equation in $$x$$?

Accepted Answer

$$ax^2 + bx + c = 0$

Question 2

Solve the quadratic equation $$x^2 - 5x + 6 = 0$ by factoring.

Accepted Answer

$$x = 2 or$$ $$x = 3$$

Question 3

Which property allows you to set each factor equal to zero when solving $$(x-4)(x+2) = 0$$?

Accepted Answer

Zero Product Principle

Question 4

Solve $$x^2 = 49$ using the square root property.

Accepted Answer

$$x = 7 or$$ $$x = -7$$

Question 5

What value should be added to $$x^2 + 8x$ to complete the square?

Accepted Answer

$$16$$

Question 6

Solve $$x^2 - 6x + 4 = 0$ by completing the square.

Accepted Answer

$$x = 3 \pm \sqrt{5}$$

Question 7

Which formula is used to solve any quadratic equation $$ax^2 + bx + c = 0$?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 8

A man’s normal systolic blood pressure, $$P$$, at age $$A$$, is modeled by $$P = 0.006A^2$ - 0.02A + 120. If$$ $$P = 125$$, what is the quadratic equation to solve for $$A$$?

Accepted Answer

$$0.006A^2 - 0.02A + 120 = 125$

Question 9

Which method is generally most efficient for solving $$x^2 - 9 = 0$?

Accepted Answer

Square Root Property

Question 10

A wheelchair ramp is 122 inches long and covers a horizontal distance of 120 inches. What is the vertical rise $$x (in $$inches)?

Accepted Answer

$$x = \sqrt{122^2$ - 120^2$}$$

Question 11

Given the quadratic equation $$2x^2 + 3x - 4 = 0$, which of the following is the correct setup for the quadratic formula?

Accepted Answer

$$x = \frac{-3 \pm \sqrt{3^2$ - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2}$$

Question 12

A piece of property is a right triangle. The longer leg is 20 m longer than twice the shorter leg. The hypotenuse is 10 m longer than the longer leg. If the shorter leg is $$x$$, what is the equation for the hypotenuse?

Accepted Answer

$$	ext{Hypotenuse} = 2x + 30$$

Question 13

Which of the following is NOT a valid method for solving all quadratic equations?

Accepted Answer

Factoring (when the equation is not factorable over the reals)

Question 14

If $$x^2 + 4x + 4 = 0$, what are the solutions?

Accepted Answer

$$x = -2$$

Question 15

Which of the following best describes the process of completing the square for $$x^2 - 10x$?

Accepted Answer

Add $$25 to $$both sides to form $$(x-5)^2$$

Question 16

Which of the following is the discriminant of the quadratic equation $$ax^2 + bx + c = 0$?

Accepted Answer

$$b^2 - 4ac$

Question 17

If the discriminant $$b^2 - 4ac$ is negative, what does this indicate about the solutions to the quadratic equation?

Accepted Answer

There are two complex (non-real) solutions.

Question 18

Which of the following real-world scenarios can be modeled by a quadratic equation?

Accepted Answer

The path of a thrown ball.

Quadratic Equations: Methods, Applications, and Problem Solving

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