Q: A quadratic equation has complex roots. Which of the following must be true about its discriminant?

The discriminant is less than zero.

Question 1

Which of the following is the correct definition of an irrational number?

Accepted Answer

A number that cannot be written as a ratio of two integers and has a non-repeating, non-terminating decimal expansion.

Question 2

A chemist measures the temperature of a reaction over time and finds that it fits the equation $T(t) = 5t + 20$, where $T$ is temperature in degrees Celsius and $t$ is time in minutes. What is the slope of the line, and what does it represent in this context?

Accepted Answer

The slope is $5$, representing the rate of temperature increase per minute.

Question 3

Which of the following equations represents a circle with center at $(2, -3)$ and radius $5$?

Accepted Answer

$(x - 2)^2 + (y + 3)^2 = 25$

Question 4

A company’s profit $P$ (in thousands of dollars) is modeled by the quadratic function $P(x) = -2x^2 + 8x + 10$, where $x$ is the number of units sold (in hundreds). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $18$ thousand dollars at $x = 2$ hundred units.

Question 5

Which of the following is the correct solution to the inequality $|2x - 3| < 7$?

Accepted Answer

$-2 < x < 5$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is $500$, which function models the population $P(t)$ after $t$ hours?

Accepted Answer

$P(t) = 500 \cdot 2^{t/3}$

Question 7

Which of the following is the correct inverse of the function $f(x) = 3x - 7$?

Accepted Answer

$f^{-1}(x) = \frac{x + 7}{3}$

Question 8

Given the quadratic equation $2x^2 - 4x - 6 = 0$, what are the solutions?

Accepted Answer

$x = 3,\ x = -1$

Question 9

A ball is thrown upward from a height of $2$ meters with an initial velocity of $10$ m/s. Its height after $t$ seconds is given by $h(t) = -4.9t^2 + 10t + 2$. After how many seconds does the ball reach its maximum height?

Accepted Answer

$t = 1.02$ seconds

Question 10

Which of the following is the correct factorization of $x^2 - 9x + 20$?

Accepted Answer

$(x - 4)(x - 5)$

Question 11

If $f(x) = x^2$ and $g(x) = x + 1$, what is $(f \circ g)(2)$?

Accepted Answer

$9$

Question 12

Which of the following describes the end behavior of the polynomial $f(x) = -2x^3 + 5x^2 - x + 7$ as $x 	o \infty$?

Accepted Answer

$f(x) 	o -\infty$

Question 13

Which of the following is the correct solution to the equation $\log_2(x) = 5$?

Accepted Answer

$x = 32$

Question 14

A radioactive substance decays according to $A(t) = 100e^{-0.2t}$, where $A$ is the amount remaining after $t$ years. How much remains after $5$ years?

Accepted Answer

$36.79$

Question 15

Which of the following is the correct solution to the system of equations: $\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$?

Accepted Answer

$x = 2,\ y = 3$

Question 16

Which of the following is the correct domain of the function $f(x) = \sqrt{5 - x}$?

Accepted Answer

$x \leq 5$

Question 17

A quadratic equation has complex roots. Which of the following must be true about its discriminant?

Accepted Answer

The discriminant is less than zero.

Question 18

Which of the following is the correct solution to the equation $2^{x+1} = 16$?

Accepted Answer

$x = 3$

Question 19

If $f(x) = \frac{1}{x-2}$, which of the following describes the vertical asymptote?

Accepted Answer

$x = 2$

Question 20

Which of the following is the sum of the roots of the equation $x^2 - 7x + 12 = 0$?

Accepted Answer

$7$

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