Q: Which of the following statements about the discriminant is correct?

If $b^2 - 4ac < 0$, the quadratic equation has two imaginary solutions.

Q: Solve for $x$ using the quadratic formula: $x^2 = 4x - 9$

$x = 2 + \sqrt{13}$ or $x = 2 - \sqrt{13}$

Q: Which step is NOT part of the process of completing the square for $ax^2 + bx + c = 0$?

Multiply both sides by the leading coefficient

Q: Given $2(x-4)^2 - 16 = 0$, what are the solutions for $x$?

$x = 4 + 2\sqrt{2}$ or $x = 4 - 2\sqrt{2}$

Q: If a quadratic equation has a discriminant of zero, what can you conclude about its solutions?

There is exactly one real solution.

Q: Which of the following is the correct quadratic formula for $ax^2 + bx + c = 0$?

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

Q: A quadratic equation models the profit $P$ (in dollars) for a company: $P = -2x^2 + 8x + 10$. What is the maximum profit, and at what value of $x$ does it occur?

Maximum profit is $26$ at $x = 2$

Question 1

Which of the following equations is a quadratic equation?

Accepted Answer

$2x^2 - 5x + 3 = 0$

Question 2

A ball is thrown upward from the ground with an initial velocity. Its height $h$ (in meters) after $t$ seconds is given by $h = -4.9t^2 + 20t$. When does the ball hit the ground again?

Accepted Answer

$t = 0$ or $t \approx 4.08$

Question 3

What is the discriminant of the quadratic equation $3x^2 - 6x + 2 = 0$ and how many real solutions does it have?

Accepted Answer

Discriminant is $(-6)^2 - 4 \cdot 3 \cdot 2 = 36 - 24 = 12$, so there are 2 real solutions.

Question 4

Solve the equation $x^2 - 9 = 0$.

Accepted Answer

$x = 3$ or $x = -3$

Question 5

Which of the following best describes the solutions to $x^2 + 4 = 0$?

Accepted Answer

Two imaginary solutions: $x = 2i$ and $x = -2i$

Question 6

A chemical reaction's rate is modeled by $r = t^2 - 6t + 8$. At what time(s) $t$ is the rate zero?

Accepted Answer

$t = 2$ or $t = 4$

Question 7

Which method is most appropriate to solve $x^2 - x - 12 = 0$?

Accepted Answer

Factoring

Question 8

Solve for $x$: $2x^2 - 1 = 9$

Accepted Answer

$x = 2$ or $x = -2$

Question 9

A medical dosage is modeled by $d = x^2 - 4x + 4$. What is the minimum dosage, and at what value of $x$ does it occur?

Accepted Answer

Minimum is $0$ at $x = 2$

Question 10

Which of the following statements about the discriminant is correct?

Accepted Answer

If $b^2 - 4ac < 0$, the quadratic equation has two imaginary solutions.

Question 11

Solve for $x$ using the quadratic formula: $x^2 = 4x - 9$

Accepted Answer

$x = 2 + \sqrt{13}$ or $x = 2 - \sqrt{13}$

Question 12

Which step is NOT part of the process of completing the square for $ax^2 + bx + c = 0$?

Accepted Answer

Multiply both sides by the leading coefficient

Question 13

A projectile's height is modeled by $h = -16t^2 + 32t + 48$. When does it reach the ground?

Accepted Answer

$t = -1$ or $t = 3$

Question 14

Given $2(x-4)^2 - 16 = 0$, what are the solutions for $x$?

Accepted Answer

$x = 4 + 2\sqrt{2}$ or $x = 4 - 2\sqrt{2}$

Question 15

If a quadratic equation has a discriminant of zero, what can you conclude about its solutions?

Accepted Answer

There is exactly one real solution.

Question 16

Solve by completing the square: $x^2 + 6x + 5 = 0$

Accepted Answer

$x = -1$ or $x = -5$

Question 17

Which of the following is the correct quadratic formula for $ax^2 + bx + c = 0$?

Accepted Answer

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

Question 18

A quadratic equation models the profit $P$ (in dollars) for a company: $P = -2x^2 + 8x + 10$. What is the maximum profit, and at what value of $x$ does it occur?

Accepted Answer

Maximum profit is $26$ at $x = 2$

Question 19

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

Accepted Answer

Factoring

Quadratic Equations: Methods of Solution and the Discriminant

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