20 medium SAT Nonlinear equations · one variable questions

Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.

Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.

Math · Advanced Math~2 per testMedium tier

None of these repeat the 25 hard Nonlinear equations · one variable questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

What is the product of the solutions to the equation $2x^{2} + 9x - 5 = 0$ ?

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Why D is right

By Vieta's formulas, the product of solutions to $2x^{2} + 9x - 5 = 0$ is $c/a = -5/2$ . Alternatively, factoring gives $(2x - 1) (x + 5) = 0$ , with solutions $x = 1/2$ and $x = -5$ , giving product (1/2)(-5) = -5/2.

Why the others are wrong

AThis extraneous value comes from using -b/a (which gives the sum of solutions) instead of c/a for the product.
BThis results from a sign error, failing to account for the negative sign in the constant term.
CThis represents only one solution (1/2) being considered, missing the contribution of the other root in the product.

Question 2Medium

$(x + 5)^{2} = (x − 3)^{2}$ . How many distinct real solutions does the given equation have?

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Why B is right

Expanding both sides gives $x^{2}$ + $10x + 25$ = $x^{2}$ − $6x + 9$ . Subtracting $x^{2}$ from both sides yields $10x + 25$ = − $6x + 9$ , which simplifies to 16x = −16, so x = −1. The equation has exactly one distinct real solution.

Why the others are wrong

AThis results from incorrectly concluding the equation has no solution after expansion.
CThis results from treating each side as producing a separate solution without recognizing the x² terms cancel, or making a sign error that preserves a quadratic.
DThis results from incorrectly concluding the equation is an identity after dropping terms during simplification.

Question 3Medium

What is the positive solution to the equation $4x^{2} + 8x - 21 = 0$ ?

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Why B is right

Using the quadratic formula x = (-b ± √ $(b^{2} - [MATH]4ac)$ [/MATH])/(2a) with $a = 4$ , $b = 8$ , and $c = -21$ , we get $\displaystyle x = (-8 \pm \sqrt{64 + 336})$ / $\displaystyle 8 = (-8 \pm \sqrt{400})/8 = (-8 \pm 20)/8$ . This yields $x = 12/8 = 3/2$ or $x = -28/8 = -7/2$ . The positive solution is $3/2$ .

Why the others are wrong

AThis value does not satisfy the original equation and represents an extraneous root.
CThis incorrectly applies a sign error to the correct positive solution.
DThis represents an incorrect calculation not derived from the quadratic formula.

Question 4Medium

$x^{2} + bx + 9 = 0$ . What value of b makes the equation have exactly one solution?

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Why B is right

For a quadratic equation to have exactly one solution, the discriminant must equal zero. For $x^{2}$ + bx + 9 = 0, the discriminant is $b^{2}$ - 4(1)(9) = $b^{2}$ - 36. Setting $b^{2}$ - 36 = 0 gives $b^{2}$ = 36, so $b = 6$ or $b = -6$ . Since the choices include only positive values, $b = 6$ .

Why the others are wrong

AThis results from taking the square root of the constant term (√9 = 3) without applying the discriminant formula.
CThis uses the constant term directly instead of solving the discriminant equation b² = 36.
DThis results from computing 4(1)(9) = 36 without taking the square root, or from incorrectly doubling the correct answer.

Question 5Medium

What is the positive solution to the equation $4x^{2} + 12x - 7 = 0$ ?

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Why A is right

Using the quadratic formula with $a = 4$ , $b = 12$ , and $c = -7$ : $\displaystyle x = (-12 \pm \sqrt{144 + 112})$ / $\displaystyle 8 = (-12 \pm \sqrt{256})/8 = (-12 \pm 16)/8$ . This gives $x = 4/8 = 1/2$ or $x = -28/8 = -7/2$ . The positive solution is 1/2.

Why the others are wrong

BThis is the correct magnitude but with the wrong sign. The positive solution is 1/2, not -1/2.
CThis results from an error in applying the quadratic formula or factoring incorrectly.
DThis is the negative solution to the equation, not the positive solution.

Question 6Medium

What is the product of the solutions to the equation $x^{2} + 9x + 14 = 0$ ?

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Why D is right

For a quadratic equation $ax^{2}$ + bx + $c = 0$ , the product of solutions is $c/a$ . Here $a = 1$ and $c = 14$ , so the product is 14/1 = 14. The equation factors as $(x + 2) (x + 7) = 0$ , confirming the product is 2 × 7 = 14.

Why the others are wrong

AThis results from incorrectly applying a negative sign to the constant term.
BThis uses the coefficient of x instead of the constant term.
CThis is the absolute value of the sum of solutions, not the product.

Question 7Medium

$(3x − 7)^{2} = 49$ . How many distinct real solutions does the given equation have?

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Why C is right

Expanding gives $9x^{2}$ − $42x + 49 = 49$ , or $9x^{2}$ − $42x = 0$ . The discriminant is (−42) $^{2}$ − 4(9)(0) = 1764. Since the discriminant is positive, the equation has exactly two distinct real solutions.

Why the others are wrong

AThis results from a sign error when computing the discriminant, incorrectly obtaining a negative value.
BThis results from incorrectly computing the discriminant as zero or recognizing only one solution when factoring.
DThis treats the simplified equation as an identity rather than a quadratic with two specific solutions.

Question 8Medium

What is the positive solution to the equation $x^{2} - 5x - 36 = 0$ ?

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Why B is right

The equation $x^{2}$ - $5x - 36 = 0$ can be factored as $(x - 9) (x + 4) = 0$ . Setting each factor equal to zero gives $x = 9$ or $x = -4$ . Since the question asks for the positive solution, the answer is 9.

Why the others are wrong

AThis results from finding the absolute value of the negative solution instead of identifying the positive solution.
CThis is the negative solution to the equation, but the question specifically asks for the positive solution.
DThis results from incorrectly applying the quadratic formula or misreading the discriminant, leading to an extraneous value that doesn't satisfy the original equation.

Question 9Medium

$9x^{2} + bx + 4 = 0$ . What value of b makes the equation have exactly one solution?

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Why C is right

For exactly one solution, the discriminant must equal zero. For $9x^{2}$ + bx + 4 = 0, the discriminant is $b^{2}$ - 4(9)(4) = $b^{2}$ - 144. Setting $b^{2}$ - 144 = 0 gives $b^{2}$ = 144, so $b = 12$ or $b = -12$ . The positive value is $b = 12$ .

Why the others are wrong

AThis results from taking the square root of the constant term (√4 = 2) without applying the discriminant formula correctly.
BThis results from computing 3(2) = 6, multiplying the square roots of the two coefficients without using the discriminant.
DThis results from computing 4(9) = 36 without taking the square root of b² = 144.

Question 10Medium

What is the product of the solutions to the equation $4x^{2} - 13x + 3 = 0$ ?

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Why D is right

For a quadratic equation $ax^{2}$ + bx + $c = 0$ , the product of solutions is $c/a$ . Here, $a = 4$ and $c = 3$ , so the product is 3/4.

Why the others are wrong

AThis incorrectly uses the coefficient b instead of the constant term c.
BThis is the constant term c without dividing by the leading coefficient a.
CThis has the wrong sign; the product c/a is positive here.

Question 11Medium

$x^{2} - 12x + 27 = 0$ What is the product of the solutions to the given equation?

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Why D is right

By Vieta's formulas, the product of solutions equals $c/a$ . For the equation $x^{2}$ - $12x + 27 = 0$ , the product is 27/1 = 27. Alternatively, factoring gives $(x - 3) (x - 9) = 0$ , so the solutions are 3 and 9, and 3 × 9 = 27.

Why the others are wrong

AThis results from a sign error, treating the constant term as negative.
BThis incorrectly uses the coefficient of x instead of the constant term.
CThis results from finding only one solution and multiplying it by an incorrect value.

Question 12Medium

$(x − 4)^{2} = 25$ . How many distinct real solutions does the given equation have?

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Why C is right

Expanding gives $x^{2}$ − $8x + 16 = 25$ , or $x^{2}$ − 8x − 9 = 0. The discriminant is (−8) $^{2}$ − 4(1)(−9) = 64 + 36 = 100. Since the discriminant is positive, the equation has exactly two distinct real solutions.

Why the others are wrong

AThis results from incorrectly expanding or combining terms, leading to an erroneous negative discriminant.
BThis results from a computational error that makes the discriminant equal zero instead of positive.
DThis treats the equation as an identity by failing to recognize it as a quadratic with finite solutions.

Question 13Medium

What is the positive solution to the equation $x^{2} - 5x = 24$ ?

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Why A is right

Rewriting the equation as $x^{2}$ - $5x - 24 = 0$ and factoring yields $(x - 8) (x + 3) = 0$ . The solutions are $x = 8$ and $x = -3$ . Since the question asks for the positive solution, the answer is 8.

Why the others are wrong

BThis is the negative solution to the equation, but the question asks for the positive solution.
CThis results from a sign error when factoring or from confusing the sum of solutions with one solution.
DThis is the sum of the absolute values of both solutions, which is extraneous to the question asked.

Question 14Medium

$25x^{2} - bx + 1 = 0$ . What value of b makes the equation have exactly one solution?

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Why C is right

For exactly one solution, the discriminant must equal zero. For $25x^{2}$ - bx + 1 = 0, the discriminant is $b^{2}$ - 4(25)(1) = $b^{2}$ - 100. Setting $b^{2}$ - 100 = 0 gives $b^{2}$ = 100, so $b = 10$ or $b = -10$ . The positive value is $b = 10$ .

Why the others are wrong

AThis results from using the constant term directly without applying the discriminant formula.
BThis results from taking the square root of the coefficient of x² (√25 = 5) instead of solving b² = 100.
DThis results from using the coefficient of x² directly instead of taking the square root of b² = 100.

Question 15Medium

What is the solution to the equation $(x + 2)^{2} = 49$ ?

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Why D is right

Taking the square root of both sides of the equation $(x + 2)^{2} = 49$ gives $x + 2 = 7$ or $x + 2 = -7$ . Solving these equations yields $x = 5$ or $x = -9$ . Both values are valid solutions.

Why the others are wrong

AThis only includes the positive solution and ignores the negative solution.
BThis only includes the negative solution and ignores the positive solution.
CThis represents the square roots of 49 without accounting for the shift by 2 in the equation.

Question 16Medium

What is the sum of the solutions to the equation $x^{2} + 8x - 9 = 0$ ?

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Why D is right

For the quadratic equation $x^{2}$ + $8x - 9 = 0$ , the sum of the solutions is $-b/a = -8/1 = -8$ . This can also be verified by factoring: $(x + 9) (x - 1) = 0$ gives solutions $x = -9$ and $x = 1$ , which sum to -8.

Why the others are wrong

AThis is the absolute value of the constant term, not the sum of solutions.
BThis results from forgetting the negative sign in the formula -b/a.
CThis is one of the individual solutions rather than the sum of both.

Question 17Medium

$(2x + 3)^{2}$ = −16. How many distinct real solutions does the given equation have?

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Why A is right

Expanding gives $4x^{2} + 12x + 9$ = −16, or $4x^{2} + 12x + 25 = 0$ . The discriminant is $12^{2}$ − 4(4)(25) = 144 − 400 = −256. Since the discriminant is negative, the equation has zero distinct real solutions.

Why the others are wrong

BThis results from dropping the negative sign on the right side or miscalculating the discriminant as zero.
CThis results from ignoring the negative right side or computing the discriminant incorrectly as positive.
DThis treats the equation as having infinitely many solutions by misunderstanding the nature of quadratic equations.

Question 18Medium

What is the positive solution to the equation $6x^{2} - 13x - 5 = 0$ ?

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Why D is right

The equation $6x^{2} - 13x - 5 = 0$ factors as $(2x - 5) (3x + 1) = 0$ , giving solutions $x = 5/2$ and $x = -1/3$ . The positive solution is 5/2.

Why the others are wrong

AThis extraneous value comes from misidentifying the absolute value of the negative solution as 1/3 instead of correctly computing it.
BThis represents only part of the positive solution, missing the correct numerator from the factorization.
CThis results from a sign error, taking the negative of the correct positive solution.

Question 19Medium

$4x^{2} - bx + 25 = 0$ . What value of b makes the equation have exactly one solution?

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Why C is right

For exactly one solution, the discriminant must equal zero. For $4x^{2}$ - bx + 25 = 0, the discriminant is $b^{2}$ - 4(4)(25) = $b^{2}$ - 400. Setting $b^{2}$ - 400 = 0 gives $b^{2}$ = 400, so $b = 20$ or $b = -20$ . The positive value is $b = 20$ .

Why the others are wrong

AThis results from taking the square root of the constant term (√25 = 5) without considering the coefficient of x².
BThis results from computing 2√25 = 10, omitting the coefficient 4 in the discriminant calculation.
DThis results from computing 4(25) = 100 without taking the square root of b² = 400.

Question 20Medium

If $x^{2} - 10x + 25 = 0$ , what is the value of x?

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Why C is right

The equation factors as $(x - 5)^{2} = 0$ . This means $x - 5 = 0$ , so $x = 5$ . This is a perfect square trinomial with a double root at $x = 5$ .

Why the others are wrong

AThis is the correct magnitude but with the wrong sign. The solution is 5, not -5.
BThis value does not satisfy the equation. Substituting x = 0 yields 25, not 0.
DThis is the coefficient of the linear term but not the solution to the equation.

What to do after medium

Medium is the tier that decides most scores. If these are landing, the hard set is where the remaining points are.

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Questions are written by SAT Climb and drawn from its own item bank. SAT® is a registered trademark of College Board, which is not affiliated with and does not endorse SAT Climb.

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What to do after medium

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