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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