PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022
How much do you know?
Quadratics
Passport to advanced math
LEARNING OBJECTIVES
After completing this chapter, you will be able to:
· Solve a quadratic equation by factoring
· Recognize the classic quadratics
· Solve a quadratic equation by completing the square
· Solve a quadratic equation by applying the quadratic formula
· Relate properties of a quadratic function to its graph and vice versa
· Solve a system of one quadratic and one linear equation
60/600 SmartPoints® (High Yield)
How much do you know?
Directions: Try the questions that follow. Show your work so that you can compare your solutions to the ones found in the Check Your Work section immediately after this question set. The “Category” heading in the explanation for each question gives the title of the lesson that covers how to solve it. If you answered the question(s) for a given lesson correctly, and if your scratchwork looks like ours, you may be able to move quickly through that lesson. If you answered incorrectly or used a different approach, you may want to take your time on that lesson.
1. If x2 + 8x = 48 and x > 0, what is the value of x − 5?
A. −9
B.−1
C.4
D. 7
2. What is the absolute value of the difference between the roots of 4x2 − 36 = 0?
A. −6
B.0
C.3
D. 6
3. Which of the following is equivalent to x2 − 6x + 10 = 0?
A. (x − 3)2 = −1
B.(x + 3)2 = 1
C.(x − 6)2 = −26
D. (x + 6)2 = 26
y2 = 2x2 − 8x + c
4. The quadratic above has only one distinct, real root. What is the value of c ?
A. −8
B.−4
C.4
D. 8
5. Which of the following equations could represent the above graph?
A. y = −x2 + 18x − 32
B.y = −x2 + 14x − 32
C.y = x2 − 14x − 32
D. y = x2 + 18x + 32
6. If (a, b) is a solution to the system of equations shown, what is the value of a, given that a > 0?
Check Your Work
1. B
Difficulty: Medium
Category: Solving Quadratics by Factoring
Strategic Advice: When finding solutions to a quadratic equation, always start by rewriting the equation to make it equal to 0 (unless both sides of the equation are already perfect squares). Then, take a peek at the answer choices—if they are all integers that are easy to work with, then factoring is probably the quickest method for solving the equation. If the answers include messy fractions or square roots, then using the quadratic formula may be a better choice.
Getting to the Answer: To make the equation equal to 0, subtract 48 from both sides to get x2 + 8x − 48 = 0. The answer choices are all integers, so factor the equation. Look for two numbers whose product is −48 and whose sum is 8. The two numbers are − 4 and 12, so the factors are (x − 4) and (x + 12). Set each factor equal to 0 and solve to find that x = 4 and x = −12. The question states that x > 0, so x must equal 4. Before selecting an answer, don’t forget to check that you answered the right question—the question asks for the value of x − 5, not just x, so the correct answer is 4 − 5 = −1. (B) is correct.
2. D
Difficulty: Hard
Category: Classic Quadratics
Getting to the Answer: Notice that this is a difference of squares, so use the formula a2 − b2 = (a + b)(a − b) with a2 = 4x2. This means that a = 2x and b2 = 36, so b = 6. Therefore, (2x + 6)(2x − 6) = 0. Set both equal to 0 and solve:
Remember that the question asks for the absolute difference of the roots. Think of a number line and count from −3 to 3. The difference is 6. Alternatively, just take the absolute value of the difference between the roots:
Thus, (D) is correct.
3. A
Difficulty: Medium
Category: Completing the Square
Getting to the Answer: The format of the answer choices makes completing the square the best approach for this question. Since the first term has a constant of 1, that makes completing the square more straightforward. Make sure to move the constant term to the right side of the equation before dividing the x term constant by 2 and squaring. Also, make sure that the squared term contains a negative constant since the x term in the original quadratic is negative:
Therefore, (A) is correct. Note that (C) is the result if you forgot to divide b by 2.
4. D
Difficulty: Medium
Category: The Quadratic Formula
Getting to the Answer: In order for the quadratic to have only one root, the discriminant must be equal to 0. Plug the numbers given into the discriminant, set it equal to 0, and solve for c:
Therefore, (D) is correct.
5. A
Difficulty: Hard
Category: Graphs of Quadratics
Strategic Advice: Use the visual information in the graph to eliminate answers quickly. A negative coefficient of the squared term means that the graph opens downward. The roots of a quadratic are where the graph crosses the x-axis.
Getting to the Answer: Because the quadratic opens downward, the x2-coefficient must be negative, so you can eliminate (C) and (D). According to the graph, the roots are x = 2 and x = 16. That means that the factored form of the quadratic will be either (−x + 2)(x − 16) or (x − 2)(−x + 16). These factored forms are actually equivalent because (−x + 2) = (−1)(x − 2) and (x − 16) = (−1)(−x + 16), which means:
(−x + 2)(x − 16) = (−1 × −1)(x − 2)(−x + 16)
(−x + 2)(x − 16) = (x − 2)(−x + 16)
Use FOIL on one of them to see if it matches the expanded form of the quadratic in (A) or (B):
(−x + 2)(x − 16) = −x2 + 16x + 2x − 32
= −x2 + 18x − 32
Thus, (A) is correct.
Another approach would be to use Picking Numbers. After eliminating (C) and (D) for having upward parabolas and calculating the roots, simply pick the more manageable of the two x-intercepts, x = 2, and plug it into the equations in (A) and (B) to see which one results in y = 0:
(A): y = −(2)2 + 18(2) − 32
y = −(4) + 36 − 32 = 0, keep.
(B): y = −(2)2 + 14(2) − 32
y = −(4) + 28 − 32 = −8 ≠ 0, eliminate.
(A) is indeed correct.
6. 5
Difficulty: Hard
Category: Systems of Quadratic and Linear Equations
Getting to the Answer: Unfortunately, this is a non-calculator question, so you’ll need to solve the system using substitution, rather than by graphing it on your calculator. Substitute the first equation for y into the second. Before you solve for x, multiply the whole equation by 2 to remove the fractions. Then, set the whole equation equal to 0 and factor:
Now, set each factor equal to 0 and solve to find that x = −1 and x = 5. The question asks only for a, which is the x- coordinate of the solution, so you do not need to substitute x back into an equation and solve for y. The two possible values of a are −1 and 5. Because the question specifies that a > 0, the answer must be 5. Grid in 5.