Q: If you graph $$y = x^2$ + 8x - 27$$, at which $$x-$$values does the graph cross the $$x-$$axis? Round to the nearest thousandth.

$$x = -9.196$$ \, \text{and} \, x = 1.196$

Question 1

Given the quadratic equation $$x^2 + 8x - 27 = 0$, which of the following is a solution to the equation, rounded to the nearest thousandth?

Accepted Answer

$$x = -9.196$$

Question 2

What is the other root of the equation $$x^2 + 8x - 27 = 0$, expressed to the nearest thousandth?

Accepted Answer

$$x = 1.196$$

Question 3

If you graph $$y = x^2$ + 8x - 27$$, at which $$x-$$values does the graph cross the $$x-$$axis? Round to the nearest thousandth.

Accepted Answer

$$x = -9.196$$ \, 	ext{and} \, x = 1.196$

Question 4

Which method would you use to find the graphical solution to $$x^2 + 8x - 27 = 0$ using a calculator?

Accepted Answer

Graph $$y = x^2$ + 8x - 27$$ and find the $$x-$$intercepts.

Question 5

Suppose you are solving $$x^2 + 8x - 27 = 0$ for a real-world scenario, such as the height of a projectile. What do the solutions x = -9.196$ and x = 1.196$ represent?

Accepted Answer

The times when the projectile is at ground level.

Question 6

Which of the following is the correct quadratic formula to solve $$x^2 + 8x - 27 = 0$?

Accepted Answer

$$x = \frac{-8 \pm \sqrt{8^2$ - 4 \cdot 1 \cdot (-27)}}{2 \cdot 1}$$

Question 7

After using the quadratic formula to solve $$x^2 + 8x - 27 = 0$, what is the value under the square root (the discriminant)?

Accepted Answer

$$196$$

Question 8

If you graph $$y = x^2$ + 8x - 27$$, what is the vertex of the parabola?

Accepted Answer

$$(-4, -43)$$

Question 9

Which of the following best describes the shape and orientation of the graph of $$y = x^2$ + 8x - 27$$?

Accepted Answer

A parabola opening upward

Question 10

If you were to factor $$x^2 + 8x - 27 = 0$, which of the following would be the correct factorization?

Accepted Answer

$$(x + 9.196)(x - 1.196) = 0$$

Question 11

If the equation $$x^2 + 8x - 27 = 0$ models the profit P$ of a company in terms of x$ units sold, what does the solution x = 1.196$ mean in context?

Accepted Answer

The company breaks even when selling approximately 1 unit.

Question 12

Which of the following is the sum of the roots of $$x^2 + 8x - 27 = 0$?

Accepted Answer

$$-8$$

Question 13

If you use a calculator to graph $$y = x^2$ + 8x - 27$$, which feature would you use to find the solutions to $$x^2 + 8x - 27 = 0$?

Accepted Answer

Zero or root function

Question 14

If $$x^2 + 8x - 27 = 0$ represents the height of a ball above the ground, which solution is physically meaningful if x$ is time in seconds?

Accepted Answer

$$x = 1.196$$

Question 15

Which of the following is the product of the roots of $$x^2 + 8x - 27 = 0$?

Accepted Answer

$$-27$$

Graphical Solutions to Quadratic Equations in College Algebra

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