# SAT CollegeBoard Practice Test (1-8) Quadratic Equation and Inequation

###### 1. (Test $1 / Q 15 /$ No Calculator)

If $(a x+2)(b x+7)=15 x^{2}+c x+14$ for all values of $x$, and $a+b=8$, what are the two possible values for $c$ ?
 A) 3 and 5 B) 6 and 35 C) 10 and 21 D) 31 and 41

###### 2. (Test1/Q 25/Calculator)

$$h=-4.9 t^{2}+25 t$$ The equation above expresses the approximate height $h$, in meters, of a ball $t$ seconds after it is launched vertically upward from the ground with an initial velocity of 25 meters per second. After approximately how many seconds will the ball hit the ground?
 A) $3.5$ B) $4.0$ C) $4.5$ D) $5.0$

###### 3. (Test $1 / \mathrm{Q} 30 /$ Calculator)

Which of the following is an equivalent form of the equation of the graph shown in the $x y$-plane above, from which the coordinates of vertex $A$ can be identified as constants in the equation?
 A) $y=(x+3)(x-5)$ B) $y=(x-3)(x+5)$ C) $y=x(x-2)-15$ D) $y=(x-1)^{2}-16$

###### 4. (Test2/Q $7 /$ Calculator)

$$y=x^{2}-6 x+8$$ The equation above represents a parabola in the $x y$-plane. Which of the following equivalent forms of the equation displays the $x$-intercepts of the parabola as constants or coefficients?
 A) $y-8=x^{2}-6 x$ B) $y+1=(x-3)^{2}$ C) $y=x(x-6)+8$ D) $y=(x-2)(x-4)$

###### 5. (Test3/Q $10 /$ No Calculator)

In the $x y$-plane, the parabola with equation $y=(x-11)^{2}$ intersects the line with equation $y=25$ at two points, $A$ and $B$. What is the length of $\overline{A B}$ ?
 A) 10 B) 12 C) 14 D) 16

###### 6. (Test $3 / Q 12 /$ No Calculator)

$$y=a(x-2)(x+4)$$ In the quadratic equation above, $a$ is a nonzero constant. The graph of the equation in the $x y$-plane is a parabola with vertex $(c, d)$. Which of the following is equal to $d$ ?
 A) $-9 a$ B) $-8 a$ C) $-5 a$ D) $-2 a$

###### 7. (Test6/Q 11/No Calculator)

The vertex of the parabola in the $x y$-plane above is $(0, c)$. Which of the following is true about the parabola with the equation $y=-a(x-b)^{2}+c$ ?
 A) The vertex is $(b, c)$ and the graph opens upward. B) The vertex is $(b, c)$ and the graph opens downward. C) The vertex is $(-b, c)$ and the graph opens upward. D) The vertex is $(-b, c)$ and the graph opens downward.

###### 8. (Test 7/Q 12/No Calculator)

The function $f$ is defined by $f(x)=(x+3)(x+1)$. The graph of $f$ in the xy-plane is a parabola. Which of the following intervals contains the $x$-coordinate of the vertex of the graph of $f$ ?
 A) $-4 < x < -3$ B) $-3 < x < 1$ C) $1 < x < 3$ D) $3 < x < 4$

###### 9. (Test 7/Q 15/No Calculator)

The expression $\frac{1}{3} x^{2}-2$ can be rewritten as $\frac{1}{3}(x-k)(x+k)$, where $k$ is a positive constant. What is the value of $k$ ?
 A) 2 B) 6 C) $\sqrt{2}$ D) $\sqrt{6}$

###### 10. (Test2/Q 13/No Calculator)

What is the sum of all values of $m$ that satisfy $2 m^{2}-16 m+8=0$ ?
 A) $-8$ B) $-4 \sqrt{3}$ C) $4 \sqrt{3}$ D) 8

###### 11. (Test3/Q 14/No Calculator)

What are the solutions to $3 x^{2}+12 x+6=0$ ?
 A) $x=-2 \pm \sqrt{2}$ B) $x=-2 \pm \frac{\sqrt{30}}{3}$ C) $x=-6 \pm \sqrt{2}$ D) $x=-6 \pm 6 \sqrt{2}$

###### 12. (Test4/Q 15/No Calculator)

$$x^{2}-\frac{k}{2} x=2 p$$ In the quadratic equation above, $k$ and $p$ are constants. What are the solutions for $x$ ?
 A) $x=\frac{k}{4} \pm \frac{\sqrt{k^{2}+2 p}}{4}$ B) $x=\frac{k}{4} \pm \frac{\sqrt{k^{2}+32 p}}{4}$ C) $x=\frac{k}{2} \pm \frac{\sqrt{k^{2}+2 p}}{2}$ D) $x=\frac{k}{2} \pm \frac{\sqrt{k^{2}+32 p}}{4}$

###### 13. (Test5/Q 3/No Calculator)

What are the solutions of the quadratic equation $4 x^{2}-8 x-12=0$ ?
 A) $x=-1$ and $x=-3$ B) $x=-1$ and $x=3$ C) $x=1$ and $x=-3$ D) $x=1$ and $x=3$

###### 14. (Test $6 / Q 20 /$ Calculator)

What is the sum of the solutions to $(x-6)(x+0.7)=0$ ?
 A) $-6.7$ B) $-5.3$ C) $5.3$ D) $6.7$

###### 15. (Test8/Q 14/Calculator)

Which of the following is a value of $x$ for which the expression $\frac{-3}{x^{2}+3 x-10}$ is undefined?
 A) $-3$ B) $-2$ C) 0 D) 2

###### 16. (Test2/Q 12/Calculator)

On January 1,2000 , there were 175,000 tons of trash in a landfill that had a capacity of 325,000 tons. Each year since then, the amount of trash in the landfill increased by 7,500 tons. If $y$ represents the time, in years, after January 1,2000 , which of the following inequalities describes the set of years where the landfill is at or above capacity?
 A) $325,000-7,500 \leq y$ B) $325,000 \leq 7,500 y$ C) $150,000 \geq 7,500 y$ D) $175,000+7,500 y \geq 325,000$

###### 17. (Test $3 / \mathrm{Q} 36 /$ Calculator)

 \begin{aligned} &y \leq-15 x+3000 \\ &y \leq 5 x \end{aligned} In the $x y$-plane, if a point with coordinates $(a, b)$ lies in the solution set of the system of inequalities above, what is the maximum possible value of $b$ ?

###### 18. (Test $4 / \mathrm{Q} 19 /$ Calculator)

If $3 p-2 \geq 1$, what is the least possible value of $3 p+2 ?$
 A) 5 B) 3 C) 2 D) 1

###### 19. (Test $4 / \mathrm{Q} 26 /$ Calculator)

Let $x$ and $y$ be numbers such that $-y < x < y$. Which of the following must be true?
I. $|x| < y$
II. $x>0$
III. $y>0$
 A) I only B) I and II only C) I and III only D) I, II, and III

###### 20. (Test1/Q 11/Calculator)

Which of the following numbers is NOT a solution of the inequality $3 x-5 \geq 4 x-3$ ?
 A) $-1$ B) $-2$ C) $-3$ D) $-5$

###### 23. (Test5/Q 13/Calculator)

$$\begin{gathered}y \leq 3 x+1 \\x-y>1\end{gathered}$$ Which of the following ordered pairs $(x, y)$ satisfies the system of inequalities above?
 A) $(-2,-1)$ B) $(-1,3)$ C) $(1,5)$ D) $(2,-1)$

###### 24. (Test5/Q 25/Calculator)

A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Among the remaining 150 participants, $p$ people chose the first picture in the set. If more than $20 \%$ of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of $p$ ?
 A) $p>0.20(300-36)$, where $p \leq 150$ B) $p>0.20(300+36)$, where $p \leq 150$ C) $p-36>0.20(300)$, where $p \leq 150$ D) $p+36>0.20(300)$, where $p \leq 150$

###### 25. (Test $6 / Q 10 /$ No Calculator)

Jaime is preparing for a bicycle race. His goal is to bicycle an average of at least 280 miles per week for 4 weeks. He bicycled 240 miles the first week, 310 miles the second week, and 320 miles the third week. Which inequality can be used to represent the number of miles, $x$, Jaime could bicycle on the 4 th week to meet his goal?
 A) $\frac{240+310+320}{3}+x \geq 280$ B) $240+310+320 \geq x(280)$ C) $\frac{240}{4}+\frac{310}{4}+\frac{320}{4}+x \geq 280$ D) $240+310+320+x \geq 4(280)$

###### 26. (Test $6 / Q 14 /$ No Calculator)

A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs $7.35$ pounds, and each container of fabric softener weighs $6.2$ pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener. Let $d$ represent the number of containers of detergent, and let $s$ represent the number of containers of fabric softener, where $d$ and $s$ are nonnegative integers. Which of the following systems of inequalities best represents this situation?
 A) $7.35 d+6.25 \leq 300$ ,$d \geq 2 s$ B) $7.35 d+6.2 s \leq 300$,$2 d \geq s$ C) $14.7 d+6.2 s \leq 300$, $d \geq 2 s$ D) $14.7 d+6.25 \leq 300$,$2 d \geq s$

###### 27. (Test $6 / \mathrm{Q} 5 /$ Calculator)

$$6 x-9 y>12$$ Which of the following inequalities is equivalent to the inequality above?
 A) $x-y>2$ B) $2 x-3 y>4$ C) $3 x-2 y>4$ D) $3 y-2 x>2$