PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022
Reflect
Quadratics
Passport to advanced math
Directions: Take a few minutes to recall what you’ve learned and what you’ve been practicing in this chapter. Consider the following questions, jot down your best answer for each one, and then compare your reflections to the expert responses on the following page. Use your level of confidence to determine what to do next.
What features in a quadratic equation should you look for to decide whether to factor, complete the square, or apply the quadratic formula?
Which constant in the vertex form of a quadratic function gives its maximum or minimum?
Which form of a quadratic equation gives its y-intercept?
Which form of a quadratic equation gives its x-intercepts, assuming the equation has two real roots?
How do you solve a system of one linear and one quadratic equation?
Expert Responses
What features in a quadratic equation should you look for to decide whether to factor, complete the square, or apply the quadratic formula?
Get the equation into standard form. If the coefficient in front of the squared term is 1, try factoring, but don’t spend longer than about 15 seconds on the attempt. If you can’t get the quadratic factored quickly, look at the coefficient on the middle term: if it is even, completing the square will be an efficient approach. Finally, the quadratic formula will work for any quadratic, no matter what the coefficients are.
Which constant in the vertex form of a quadratic function gives its maximum or minimum?
The vertex form is y = a(x − h)2+ k.The constant k is the y -value at the vertex, which occurs at the maximum or minimum.
Which form of a quadratic equation gives its y-intercept?
The standard form, y = ax2 + bx + c. The y -intercept is given by c .
Which form of a quadratic equation gives its x-intercepts, assuming the equation has two real roots?
The factored form, y = a (x − m) (x − n). The x -intercepts are at x= m and x= n .
How do you solve a system of one linear and one quadratic equation?
Put the linear equation in the form y = mx + b and the quadratic in the form y = ax2 + bx + c. Set the right sides of the equations equal to each other and solve.
Next Steps
If you answered most questions correctly in the “How Much Have You Learned?” section, and if your responses to the Reflect questions were similar to those of the PSAT expert, then consider Quadratics an area of strength and move on to the next chapter. Come back to this topic periodically to prevent yourself from getting rusty.
If you don’t yet feel confident, review those parts of this chapter that you have not yet mastered. In particular, study the table describing the different forms of quadratics in the Graphs of Quadratics lesson. Then, try the questions you missed again. As always, be sure to review the explanations closely.
Answers and Explanations
1. B
Difficulty: Easy
Getting to the Answer: FOIL the binomials 
First: (x)(x) = x2. Outer: (2)(x). Inner: (−4)(x). Last: (2)(−4) = −8. Add all the terms together and combine like terms: x2 − 4x + 2x − 8 = x2 − 2x − 8. The correct answer is (B).
2. 0
Difficulty: Easy
Getting to the Answer: All the question is really asking you to do is solve for the zeros and subtract them:
The quadratic has only one unique solution, 4, so the positive difference between the zeros of the function is 0. Grid in 0.
3. A
Difficulty: Easy
Getting to the Answer: Rearrange the equation first so you can factor 4 out. From there, divide by 4, then factor as usual:
Keep in mind that while z is equal to −10 or 2, the question asks only for the positive value, which is (A). Reading carefully, you could have eliminated (B) before doing any calculations because it includes a negative value.
4. C
Difficulty: Medium
Strategic Advice: None of the choices has a remainder, suggesting that you probably will not need polynomial division for this question. Try factoring the numerator and denominator to see if something will cancel out.
Getting to the Answer: Start by factoring out a 2 in the denominator to make that quadratic a bit simpler. Once there, factor to reveal an (x − 2) term that will cancel out:
Unfortunately, none of the answer choices match. Try factoring −1 out of the numerator:
The correct answer is (C).
5. B
Difficulty: Hard
Getting to the Answer: Set each answer choice equal to zero and factor to determine which one meets the criteria posed in the question: the solutions must have different signs and the positive solution must have a greater absolute value than the negative solution.
(A): 
The solutions are 2 and −3. They have different signs, but the negative solution has the greater absolute value. Eliminate (A).
(B): 
This time, the solutions are −2 and 3, so the criteria are met. (B) is correct.
For the record:
(C): 
This has the same solutions as (A). Eliminate (C).
(D): 
The solutions are 0 and 2. Eliminate (D) and confirm that (B) is correct.
6. D
Difficulty: Medium
Getting to the Answer: Expand the first factor to take advantage of the difference of squares, and then use FOIL to multiply the two factors that remain:
This matches (D).
7. A
Difficulty: Medium
Getting to the Answer: This question can be solved two ways. One is to factor each choice and solve for the solutions, and whichever does not have a solution of x = −4 is correct. Another way is to plug in x = −4 for each choice; if the equation does not equal 0, then that choice is correct. Luckily, plugging x = −4 into the first choice results in 64 and not 0. Choice (A) factored would be (x − 4)2, so its only unique root is x = 4. Hence, (A) is correct.
8. B
Difficulty: Medium
Getting to the Answer: Start by noticing that the area equation is a difference of perfect squares. Use the difference of squares formula (a2 − b2) = (a + b)(a − b) where (x4 − 196) = (x2 + 14)(x2 − 14). Because area is length times width (A = lw) and the width is x2 − 14, the length must be x2 + 14, so (B) is correct.
9. D
Difficulty: Medium
Getting to the Answer: The volume equation for a rectangular prism is V = lwh, so work with one variable at a time and then plug all of the dimensions in. The height will just be x, so h = x. The width is 7 less than the height, so subtract 7 from the height, or x, which gives w = x − 7. The length is 14 more than the width, so l = (x − 7) + 14 = x + 7. Plug all of these values into the volume formula and recognize a difference of squares to get x(x − 7)(x + 7) = x(x2 − 49). Thus, (D) is correct.
10. B
Difficulty: Medium
Getting to the Answer: Factor out the common term p2 to get p2(16m2 + 72 mz + 81z2). Eliminate (D), which is equivalent until the final term. Notice that 16m2 + 72 mz + 81z2 is a classic quadratic:
16m2 + 72 mz + 81z2 = (4m + 9z)2
So the original expression is equal to p2(4m + 9z)2. Thus, (B) is correct. Another way to solve would have been to FOIL, or distribute, each choice and see which one results in the given expression.
11. C
Difficulty: Medium
Strategic Advice: Equations that are equivalent have the same solutions, so you are looking for the equation that is simply written in a different form. You could expand each of the equations in the answer choices, but unless you get lucky, this strategy will use up quite a bit of time. The answer choices are written in vertex form, so use the method of completing the square to rewrite the equation given in the question stem.
Getting to the Answer: First, subtract the constant, 17, from both sides of the equation. To complete the square on the right-hand side, find 
and add the result to both sides of the equation:
Next, factor the right-hand side of the equation (which should be a perfect square trinomial), and rewrite it as a square. Finally, solve for y:
This matches (C).
12. B
Difficulty: Medium
Getting to the Answer: Since the first term has a constant of 1, completing the square can be used right away. Make sure to move the constant term to the right side of the equation before dividing the x term by 2 and squaring:
Therefore, (B) is correct.
13. D
Difficulty: Medium
Getting to the Answer: Since the first term has a constant of 1, completing the square can be used right away. Make sure to move the constant term to the right side of the equation before dividing the x term constant by 2 and squaring:
Therefore, (D) is correct.
14. A
Difficulty: Hard
Getting to the Answer: When something doesn’t factor cleanly, consider completing the square or using the quadratic formula. Start by dividing the entire equation by 4 so that the x2-coefficient is 1:
Factoring won’t work here. The coefficient b is even, so try completing the square:
The question asks for just one solution, so (A) is the correct answer.
15. D
Difficulty: Easy
Getting to the Answer: There are no zeros for choice (D) because the parabola never intersects or touches the x-axis. The discriminant of the quadratic formula for choice (D) will be negative because there are no real zeros.
There is a pair of zeros for choices (A) and (B). There is one zero for choice (C).
16. D
Difficulty: Medium
Strategic Advice: When factoring isn’t easy, try a different approach. If you were able to use a calculator, the fastest method might be to graph the function. Because this is a no-calculator question, use the quadratic formula.
Getting to the Answer: The first step in answering the question is to manipulate the equation so that it’s equal to 0:
Now, solve using the quadratic formula:
When you solve using the quadratic formula, you get a negative number under the square root, which means there are no real solutions. The correct answer is (D).
Although you couldn’t use a graphing calculator for this question, for the record, if you could graph the function, you’d see that the graph does not cross the x-axis, which means no real solutions:
(D) is indeed correct.
17. A
Difficulty: Hard
Strategic Advice: The roots of an equation are the same as its solutions. The equation doesn’t factor using reverse FOIL, so you’ll have to use a different method to find the solutions. The equation is already written in the form y = ax2 + bx + c, and the coefficients are fairly small, so using the quadratic formula is probably the quickest method.
Getting to the Answer: Note the values that you’ll need: a = 1, b = 8, and c = −3. Then, substitute these values into the quadratic formula and simplify:
This is not one of the answer choices, which tells you that you’ll need to simplify the radical, but before you do, you can eliminate (C) and (D) because the non-radical part of the solutions is 
not 4. To simplify the radical, look for a perfect square that divides into 76 and take its square root:
This matches (A).
18. B
Difficulty: Medium
Getting to the Answer: The discriminant is part of the quadratic formula. When the discriminant is negative, there are no real solutions to the equation because the square root of a negative number is not a real number. When the discriminant is 0, there is only one solution. For there to be two real solutions, the discriminant, b2 − 4 ac, must be greater than 0.
(A): a = 2, b = 4, c = 2. The discriminant is 42 − 4(2)(2) = 16 − 16 = 0, so this has one real solution. Eliminate (A).
(B): a = 5, b = 5, c = −5. The discriminant is 52 − (4)(5)(−5) = 25 + 100 = 125, so choice (B) must have two real solutions and is correct.
For the record, the discriminant of (C) is −75 and the discriminant of (D) is 0.
19. A
Difficulty: Medium
Getting to the Answer: Since the answer choices have radicals, factoring the equation would be extremely difficult. Use the quadratic formula to find the roots of the equation:
This matches (A).
20. D
Difficulty: Medium
Strategic Advice: Quadratic equations can be written in several forms, each of which reveals something important about the graph. For example, the vertex form of a quadratic equation, y = a(x − h)2 + k, gives the minimum or maximum y-value of the function, k, while the standard form, y = ax2 + bx + c, shows the y- intercept, c.
Getting to the Answer: The factored form of a quadratic equation makes it easiest to calculate the solutions to the equation, which graphically represent the x- intercepts. Choice (D) is the only equation written in factored form and therefore must be correct. You can set each factor equal to 0 and quickly solve to find that the x- intercepts of the graph are 
and x = 2, which agree with the graph. (Note that each unit on the graph is 0.5.)
21. B
Difficulty: Medium
Getting to the Answer: The parabola opens downward, so the a term should be negative. Unfortunately, all the choices have a negative a term, so you’re not able to eliminate any of them. All of the choices appear to be in vertex form, so take a look at the graph: the vertex appears to be at (3, 2); only (A) and (B) match this. To decide between them, plug in some values from points on the graph, such as the two x-intercepts, (2, 0) and (4, 0). Check if f (2) = 0 for (A): −((2) − 3)2 + 2 = −(−1)2 + 2 = −(1) + 2 = 1. Eliminate (A) because f (2) ≠ 0. Only (B) is left and is correct. For this equation, both x = 2 and x = 4 produce a result of f(x) = 0.
22. B
Difficulty: Medium
Getting to the Answer: According to the graph, one x- intercept is to the left of the y- axis and the other is to the right. This tells you that one x- intercept has a positive x-value and the other x- intercept has a negative x-value, so you can immediately eliminate choices (A) and (C) because both factors have the same sign. To choose between choices (B) and (D), find the x- intercepts by setting each factor equal to 0 and solving for x. In choice (B), the x- intercepts are 7 and −3. In choice (D), the x- intercepts are 1 and −10. Choice (B) is correct because the x- intercepts are exactly 10 units apart, while the x- intercepts in choice (D) are 11 units apart. Alternatively, given that the distance between the two intercepts is 10 units, you could have just found the choice with the two factors for which the positive difference between the numerical terms is 10: 3 −(−7) = 10.
23. B
Difficulty: Medium
Strategic Advice: The coefficients are given as unknowns, so you’ll need to think about how their values affect the graph. You’ll need to recall certain vocabulary. Recall that increasing means rising from left to right, while decreasing means falling from left to right, and zero is another way of saying x- intercept. Compare each statement to the graph to determine whether it is true, eliminating choices as you go. Remember, you are looking for the statement that is NOT true.
Getting to the Answer: The parabola opens downward, so a must be negative, which means you can eliminate (A). When a quadratic equation is written in standard form, c is the y-intercept of the parabola. According to the graph, the y- intercept is above the x- axis and is therefore positive, so the statement in (B) is false, making it the correct answer.
For the record, (C) is true because the graph rises from left to right until you get to x = 3, and then it falls. Choice (D) is true because the zeros are the same as the x-intercepts, and the graph does intersect the x- axis at −2 and 8.
24. A
Difficulty: Hard
Getting to the Answer: Begin by considering the shape of the parabola formed by the path of the ball: because the ball starts and ends at ground level (y = 0) and travels a horizontal distance of 150 feet (x = 150), it must have x-intercepts of (0, 0) and (150, 0). In addition, the question tells you that it reaches a maximum height of 45 feet (y = 45) and, because the vertex is halfway between the two x-intercepts, the vertex must be at (75, 45). The vertex is above the x-intercepts, so this is a downward parabola. You can immediately eliminate choices (C) and (D), which have positive a values, making them upward parabolas. To decide between (A) and (B), try plugging in the coordinates of the vertex or an x-intercept to see which equation holds. For example, here are the calculations if you use (150, 0):
(A): 0 = −0.008(150)2 + 1.2(150) = −180 + 180 = 0, keep.
(B): 0 = −0.008(150)2 − 150(150) = −180 − 22,500 = −22,680 ≠ 0, eliminate.
Thus, (A) is correct.
If you have time, you could also graph each equation in your graphing calculator and find the one that has a maximum value of 45, which would show again that (A) is the only equation for which this is true.
25. D
Difficulty: Hard
Getting to the Answer: Make sure you read the axis labels, the question, and the answer choices carefully. The vertical axis is labeled in feet, while the horizontal axis is labeled in yards. The answer choices are given in feet, so you’ll need to convert the yards to feet.
The question asks for the difference between the horizontal and vertical distances the cannonball travels. You have a parabola-shaped graph, so sketch in a quadratic model, making sure to extend it past the last point all the way back to the x-axis.
Horizontally, the cannonball starts at 0 yards and travels to about 485 yards, or 485 × 3 = 1,455 feet. Determining vertical travel is a bit more involved. According to the graph, the cannonball’s peak height is about 75 feet, but it started at 10 vertical feet (not 0), making the net upward distance traveled 65 feet. Subtract to find the difference, 1,455 − 65 = 1,390 feet, which is (D).
Note: Don’t worry if you didn’t draw the model exactly right or if you didn’t get the exact same answer. The choices should be far enough apart that you’ll still know which one is correct.
26. B
Difficulty: Medium
Getting to the Answer: Even though one of the equations in this system is not linear, you can still solve the system using substitution. You already know that y is equal to 3x, so substitute 3x for y in the second equation. Don’t forget that when you square 3x, you must square both the coefficient and the variable:
The question asks for the value of x2, not x, so there is no need to take the square root of 36 to find the value of x. Choice (B) is correct.
27. D
Difficulty: Hard
Strategic Advice: Solving each system would be absurdly time-consuming. Backsolving will be faster.
Getting to the Answer: Substitute x = −8, the x-value at one of the points of intersection, into each equation. The correct system will have both equations equaling the same number, which is the y-value at that point of intersection. Look at each answer choice:
(A): 
7 ≠ 253, so eliminate (A).
(B): 
0 ≠ 17, so eliminate (B).
(C): 
0 ≠ 23, so eliminate (C).
(D): 
7 = 7, so the two equations in (D) intersect at the point (−8, 7). Thus, (D) is correct.
28. D
Difficulty: Hard
Strategic Advice: All of the choices present a system of equations that includes a parabola and a line. For a system of equations to have only one solution, the graphs must have only one point of intersection.
Getting to the Answer: Notice that all of the choices have the same parabola with a vertex of (−3, −5). Choices (B) and (D) have horizontal lines, which are easy to check. Choice (B) has a horizontal line at y = 5, but with the parabola facing downward, the parabola will never intersect it. Choice (D) has a horizontal line at y = −5, which is where the parabola’s vertex lies. This means the line is tangent to the parabola and will touch the parabola only once to create only one solution. Thus, (D) is correct. For the record, (A) has two points of intersection and (C) has none.
29. C
Difficulty: Hard
Category: Systems of Quadratic and Linear Equations
Getting to the Answer: If this question is in the calculator section, you can graph the two functions simultaneously and observe that they intersect each other twice. If you are not visually sold on what the correct answer is, you can do a little investigating by using your calculator to tell you the values of the vertex (the minimum) and the points of intersection.
If this is in the non-calculator section, you’ll instead need to solve this by hand by setting the two equations equal to each other to see whether they intersect and, if so, how many times:
You’ll see that the functions do intersect each other at two locations, x = −1 and x = 2. You now need to determine the location of the vertex to compare. Using the formula given for the x-coordinate of the vertex (h), you can do this quite easily:
You need not calculate the y-coordinate (k) because you already have the answer. Both of the points of intersection, −1 and 2, occur to the right of the vertex. The answer is (C).
30. B
Difficulty: Medium
Category: Graphs of Quadratics
Getting to the Answer: Set f(x) equal to g(x): 2x2 − 5 = 6x2 − 7. Isolate the x2 terms on one side to get 2 = 4x2, so x2 = 
Take the square root of both sides to see that 
which means that the two intersections of the functions occur when 
and 
. None of the choices match, so multiply the numerator and denominator by 
to convert these to 
. From the graph, you can see that these are the values of ± z, so 
. (B) is correct.
Alternatively, you could plug in the coordinates of one of the intersections into either function. Using f(x), the y-coordinate is −4 and the x-coordinate is z. So, −4 = 2z2 − 5. The math works out exactly the same as for the first approach: z2 = 
, so 
, and, from the graph, you can determine that 
. Again, (B) is correct.
31. B
Difficulty: Easy
Category: Graphs of Quadratics
Getting to the Answer: Sketch a regression line on the graph. You’ll notice it has a slight curve. Extending the regression line to the x-axis allows you to reasonably estimate what your fuel economy at 100 miles per hour (mph) will be. Notice that the miles per gallon (mpg) drop from 80 mph to 90 mph is slightly less than 10, so it is reasonable to expect a drop of another full 10 mpg as the curve steepens from 90 mph to 100 mph. Therefore, the mpg at 100 mph would be about 10, so (B) is correct.
32. 5
Difficulty: Easy
Category: Solving Quadratics by Factoring
Getting to the Answer: Factor the given quadratic by finding factors of 20 that add, or subtract, to get the middle constant 9. Notice that −4 and −5 will multiply to get 20 and add to get −9. So, x2 − 9x + 20 = (x − 4)(x − 5). Set both equal to 0 and solve for x to get x = 4 and x = 5. Remember that the question asks for m, where m > n, or in other words for the larger of the two roots, so grid in 5.
33. D
Difficulty: Hard
Category: Graphs of Quadratics
Getting to the Answer: Look for an equation among the choices that has a maximum value that is less than the maximum height of Shawna’s toss. To determine the peak height of Shawna’s throw, convert the given equation to vertex form, y = a(x − h)2 + k, where the maximum value is given by k. Notice that the polynomial within the parentheses factors to(t − 2)2. Thus, you can restate the given equation as −5(t − 2)2 + 22. So the vertex of the equation for Shawna’s throw is (2, 22), which means that the maximum height was 22 meters.
Conveniently, the choices are all stated in vertex form. The only one with the k term less than 22 is (D), which makes that the correct choice. (Notice that this equation differs from the restated version of the given equation only by the k term.)
34. C
Difficulty: Medium
Category: Graphs of Quadratics
Getting to the Answer: When a quadratic equation is written in vertex form, y = a(x − h)2 + k, the minimum value (or the maximum value if a < 0) is given by k, and the axis of symmetry is given by the equation x = h. The question states that the minimum of the parabola is −3, so look for an equation where k = −3. You can eliminate choices (A) and (B) because k = 2 in both equations. The question also states that the axis of symmetry is x = 2, so h must be 2. Be careful: this can be tricky. The equation in choice (D) is not correct because the vertex form of a parabola includes the term (x − h) not (x + h), so (x + 2) should be interpreted as (x − (−2)), with axis of symmetry at x = −2. This means (C) is correct.
35. C
Difficulty: Medium
Category: Solving Quadratics by Factoring
Getting to the Answer: Rearrange the equation into the standard quadratic form by subtracting everything from the right side of the equal sign: 
. Next, divide the equation by 2 to make factoring easier: 
. Now, use reverse FOIL to determine the factors. You need factors of 12 that sum to −7. Those are −3 and −4, so your equation factors to (x − 3)(x − 4) = 0. That means the solutions to the equation are x = 3 and x = 4. Since only one of these is among the answers, (C) is correct.
36. C
Difficulty: Hard
Category: Completing the Square
Getting to the Answer: First divide both sides by 4, so that 4x2 − 8x + 64 = 0 becomes x2 − 2x + 16 =0. Notice that this cannot be solved by factoring, so use the completing the square method:
Therefore, (C) is correct.
37. A
Difficulty: Medium
Category: Classic Quadratics
Getting to the Answer: Notice that this quadratic is a difference of squares, so use the formula a2 − b2 = (a + b)(a − b). Set a2 equal to 64x2 and take the square root of both sides to find that a = 8x. Do the same thing with b2 and 81y2 to find that b = 9y. Fill in (a + b)(a − b) with the values you found for a and b to get (8x + 9y)(8x − 9y). Therefore, (A) is correct.
38. D
Difficulty: Hard
Category: Quadratic Formula
Strategic Advice: The discriminant is the part of the quadratic formula that determines whether a quadratic equation has 1 or 2 distinct real solutions or only imaginary solutions. Note that a quadratic will have only one distinct real solution when the discriminant equals 0.
Getting to the Answer: Convert the equations to standard quadratic form, if necessary, and calculate the discriminant for each choice to see which one equals 0:
(A) converts to 4x2 − 3x + 8 = 0. The discriminant is (−3)2 − 4(4)(8) = 9 − 128 = −119. This is not equal to 0, so eliminate (A).
(B) converts to x2 + 10x − 2 = 0. The discriminant is (10)2 − 4(1)(−2) = 100 + 8 = 108. Eliminate (B).
(C) is already in the proper form. The discriminant is (2)2 − 4(7)(−5) = 4 + 140 = 144. Eliminate (C).
Only (D) is left, so it is correct. For the record, the discriminant for (D) does, in fact, equal 0. Divide through by the common factor of 3 to get x2 − 2x + 1 = 0, so the discriminant is (−2)2 −4(1)(1) = 0.
39. 9
Difficulty: Hard
Category: Systems of Quadratic and Linear Equations
Getting to the Answer: Rearrange both equations to isolate y in terms of x:
Set both equations equal to each other to find where they intersect. Combine like terms on one side of the equation:
This quadratic equation can be solved by factoring since two factors of −14, 7 and −2, add to 5. Hence, (x + 7)(x − 2) = 0 and x = −7 and x = 2. The question asks for the absolute difference, so subtract the roots and evaluate as an absolute value:
Grid in 9.
40. A
Difficulty: Hard
Category: Completing the Square
Strategic Advice: Vertex form is usually a good option when questions ask for coordinates of the vertex. One of the best methods to convert quadratic equations into vertex form is completing the square.
Getting to the Answer: The x2 term has a coefficient of 1, so no manipulation of the equation is necessary before completing the square. Move the constant term to the right side of the equation before dividing the x term’s coefficient by 2 and squaring:
Now the equation y = (x − 7)2 − 46 is in vertex form, y = a(x − h)2 + k, where the vertex is (h, k). To find the x-coordinate of the vertex, or h, be aware of the negative sign before h in vertex form. In this case, −h = −7, so h = 7. Only (A) has an x-coordinate of 7, so it must be correct.
For the record, the y-coordinate, or k, of the vertex is −46. The vertex therefore is (h, k) = (7, −46), so (A) is indeed correct.