Quadratics questions on Digital SAT Math reward knowing a handful of shortcuts cold: the sum and product of a quadratic's roots can be read directly from its coefficients (sum = −b/a, product = c/a) without ever solving for the roots themselves, and the discriminant (b² − 4ac) instantly tells you whether an equation has zero, one, or two real solutions — which is exactly what "exactly one solution" and "no real solutions" problems are testing. Vertex form, factoring, and real-world revenue and projectile-motion models round out the rest of the topic.
The 21 problems below cover this full range, including several discriminant-based "exactly one solution" and "no real solutions" problems, vertex-form reasoning, rational equations that reduce to quadratics, and real-world interpretation questions. Every problem includes a complete step-by-step explanation, hidden by default so you can test yourself honestly before checking your work. Whenever a problem asks for the sum or product of a quadratic's solutions specifically (rather than the solutions themselves), reach for −b/a and c/a first — it's almost always faster than solving the equation directly. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
4x² + kx + 9 = 0. In the given equation, k is a positive constant. The equation has exactly one real solution. What is the value of k?
A quadratic has exactly one real solution when its discriminant equals zero: b² − 4ac = 0. Here, a = 4, b = k, c = 9: k² − 4(4)(9) = 0.
Solve: k² = 144, so k = ±12. Since k is positive, k = 12.
x² − x − 12 = 0. What is the sum of the solutions to the given equation?
For any quadratic ax² + bx + c = 0, the sum of the solutions equals −b/a without needing to solve for the roots individually. Here, a = 1 and b = −1: sum = −(−1)/1 = 1.
Check by factoring: x² − x − 12 = (x − 4)(x + 3), giving roots 4 and −3, which do sum to 1. The correct answer is C.
3x² − 2x − 7 = 0. What is the sum of the solutions to the given equation?
The sum of the solutions to ax² + bx + c = 0 equals −b/a. Here, a = 3 and b = −2.
Sum = −(−2)/3 = 2/3.
h(t) = −4.9t² + 10t. The function h models the height h(t), in meters, of a football t seconds after it is kicked. What is the interpretation of h(2) = 0.40 in this context?
In h(t), the input t is measured in seconds, and the output is the height in meters — not necessarily the maximum height, just the height at that specific moment.
So h(2) = 0.40 means 2 seconds after the kick, the football's height is 0.40 meter. The correct answer is C.
18x² + 24x + c = 0. In the given equation, c is a constant. The equation has exactly one solution. What is the value of c?
Set the discriminant equal to zero: 24² − 4(18)(c) = 0, so 576 − 72c = 0.
Solve: 72c = 576, so c = 8.
The function g is a quadratic function. In the xy-plane, the graph of y = g(x) has a vertex at (−1, −4) and passes through the points (−2, −43) and (1, −160). What is the value of g(0) − g(2)?
Write g in vertex form: g(x) = a(x + 1)² − 4. Use (−2, −43): a(−1)² − 4 = a − 4 = −43, so a = −39. Check with (1, −160): −39(2)² − 4 = −156 − 4 = −160 ✓.
So g(x) = −39(x+1)² − 4. Then g(0) = −39(1) − 4 = −43, and g(2) = −39(9) − 4 = −355. g(0) − g(2) = −43 − (−355) = 312. The correct answer is D.
An auditorium has seats for 3,200 people. Tickets to attend a show at the auditorium currently cost $8.00. For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation −100x² + 2,400x = 25600, where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If the equation is graphed in the xy-plane, at what value of x is the maximum of the graph?
The maximum of a downward-opening parabola y = ax² + bx occurs at its vertex, x = −b/(2a), regardless of any specific point the graph passes through. Here, a = −100 and b = 2,400.
x = −2,400/(2 × −100) = −2,400/−200 = 12. The correct answer is B.
(x + 1)/(5x²) = k/x. In the given equation, k is a constant. The solution to the given equation is 1/174. What is the value of k?
Cross-multiply: x(x + 1) = 5x²k. Divide both sides by x (valid since x ≠ 0): x + 1 = 5xk, so k = (x + 1)/(5x).
Substitute x = 1/174: k = (1/174 + 1)/(5 × 1/174) = (175/174)/(5/174) = 175/5 = 35.
If (x + 3)² = 30, what is the value of x² + 6x?
Expand the left side: x² + 6x + 9 = 30.
Subtract 9 from both sides: x² + 6x = 21. The correct answer is A.
What is the solution to the equation 2√x = 3?
Isolate the radical: √x = 3/2.
Square both sides: x = (3/2)² = 9/4. The correct answer is C.
3x² − 4x + 7 = 0. How many distinct real solutions does the given equation have?
Check the discriminant: b² − 4ac = (−4)² − 4(3)(7) = 16 − 84 = −68.
Since the discriminant is negative, the equation has no real solutions. The correct answer is A.
x² − 4x − 1 = 0. A solution to the given equation is x = √k + 2. What is the value of k?
Use the quadratic formula: x = (4 ± √(16 + 4))/2 = (4 ± √20)/2 = 2 ± √5.
Matching the positive-root solution 2 + √5 to the given form √k + 2, we get √k = √5, so k = 5.
0.24x² + 0.56x + 1.52. The given expression can be rewritten as a(3x² + 7x + 19), where a is a constant. What is the value of a?
Match the x² coefficients: 0.24 = 3a, so a = 0.08.
Check against the other terms: 7(0.08) = 0.56 ✓ and 19(0.08) = 1.52 ✓, confirming a = 0.08.
In the equation 10x² + 50x + c = 0, c is a constant. If the equation has exactly one solution, what is the value of c?
Set the discriminant equal to zero: 50² − 4(10)(c) = 0, so 2,500 − 40c = 0.
Solve: 40c = 2,500, so c = 125/2. The correct answer is D.
The solutions to x² + 10x + 23 = 0 are r and s, where r < s. The solutions to x² + 10x + 17 = 0 are t and u, where t < u. The solutions to x² + 20x + c = 0, where c is a constant, are r + u and s + t. What is the value of c?
Solve each quadratic: x²+10x+23=0 gives x = −5 ± √2, so r = −5−√2, s = −5+√2. x²+10x+17=0 gives x = −5 ± 2√2, so t = −5−2√2, u = −5+2√2.
Compute r+u = (−5−√2)+(−5+2√2) = −10+√2, and s+t = (−5+√2)+(−5−2√2) = −10−√2. For x²+20x+c=0, c equals the product of the roots: (−10+√2)(−10−√2) = 100 − 2 = 98.
1/x + 1/(x−3) = 6/(x²−3x). What are all the solutions to the given equation?
Factor the right side's denominator: x²−3x = x(x−3), which is the common denominator. Combine the left side: [(x−3)+x]/[x(x−3)] = 6/[x(x−3)].
Since the denominators match, set numerators equal: 2x − 3 = 6, so x = 9/2. This value doesn't violate the domain restrictions (x ≠ 0, 3). The correct answer is D.
x² − 12x + c = 0. In the given equation, c is a constant. The equation has no real solutions if c > n. What is the least possible value of n?
No real solutions requires a negative discriminant: (−12)² − 4c < 0, so 144 − 4c < 0.
Solve: 144 < 4c, so c > 36. Since the equation has no real solutions exactly when c > 36, the least possible value of n is 36. The correct answer is D.
22x² + (22s + r)x + rs = 0. In the given equation, r and s are positive constants. The product of the solutions to the given equation is krs, where k is a constant. What is the value of k?
For ax² + bx + c = 0, the product of the solutions equals c/a. Here, the constant term is rs and the leading coefficient is 22, so the product of the solutions is rs/22.
Setting this equal to krs: krs = rs/22, so k = 1/22.
y = −4x² − 23 and y = qx − 19. In the given system of equations, q is a positive constant. The graphs of the equations in the given system intersect at exactly one point (x, y) in the xy-plane. What is the value of x?
Set the equations equal: −4x² − 23 = qx − 19, which rearranges to 4x² + qx + 4 = 0. For exactly one intersection point, the discriminant must equal zero: q² − 4(4)(4) = 0, so q² = 64, giving q = 8 (positive).
Substitute back: 4x² + 8x + 4 = 0. Divide by 4: x² + 2x + 1 = 0, which factors as (x+1)² = 0, giving x = −1. The correct answer is D.
y = −(1/4)x² + x + 28. The given equation models a company's active projects over 6 months, where y is the estimated number of active projects x months after the end of April 2012, where 0 ≤ x ≤ 6. Which statement is the best interpretation of the y-intercept of the graph of this equation in the xy-plane?
The y-intercept occurs at x = 0: y = −(1/4)(0)² + 0 + 28 = 28. Since x represents months after the end of April 2012, x = 0 corresponds to the end of April 2012 itself.
So the y-intercept means the estimated number of active projects was 28 at the end of April 2012. The correct answer is B.
One of the factors of 4x³ + 88x² + 468x is x + b, where b is a positive constant. What is the smallest possible value of b?
Factor out the common term first: 4x³ + 88x² + 468x = 4x(x² + 22x + 117). Now factor the quadratic: look for two numbers multiplying to 117 and adding to 22 — that's 9 and 13, since 117 = 9 × 13.
So the full factorization is 4x(x+9)(x+13). The two constants that fit the form x+b are 9 and 13, and the smallest is 9.
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