Quadratic Equation −3x² − 5x + 6 = 0 — SAT Math Answer
Answer First: Solving −3x² − 5x + 6 = 0 on the Digital SAT
The correct answer to this Quadratic Equation −3x² − 5x + 6 = 0 — SAT Math problem is −5/6 + √97/6. This is the greatest solution to the quadratic, found by applying the quadratic formula and carefully handling the negative leading coefficient.
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This question tests a core skill in the Passport to Advanced Math domain: solving quadratic equations where the coefficient of x² is negative. The College Board regularly includes these problems on the Digital SAT to check whether students can manage sign changes and simplification without error. According to official SAT skill breakdowns, Passport to Advanced Math accounts for roughly 7-10 questions across both math modules, and solving quadratics is one of the most frequently tested sub-skills.
Many students lose points on this question not because they do not know the quadratic formula, but because they mishandle the negative signs when plugging in a, b, and c. The problem looks straightforward, but the negative denominator creates a trap that flips the expected order of solutions.
Core Skill: Solving Quadratics with Negative Leading Coefficients
The quadratic formula x = [−b ± √(b² − 4ac)] / (2a) works for any quadratic in standard form ax² + bx + c = 0. When a is negative, two things happen that can trip you up:
That negative coefficient in −3x² − 5x + 6 = 0 is a classic SAT trap. Start with free lesson notes and Targeted Practice, no card needed.
Start 1-day free trial →- The denominator 2a becomes negative, which flips the relative size of the two solutions.
- The term −b may change sign depending on whether b itself is negative.
In this equation, a = −3, b = −5, and c = 6. The formula becomes x = [−(−5) ± √((−5)² − 4(−3)(6))] / (2(−3)). Simplifying step by step: −(−5) becomes +5, (−5)² becomes +25, and −4(−3)(6) becomes +72. Inside the square root, you get 25 + 72 = 97. The denominator is 2(−3) = −6. So x = [5 ± √97] / (−6).
Now the critical insight: because the denominator is negative, the two solutions are not in the order you might expect. Normally, (5 + √97) / (−6) is the smaller solution, and (5 − √97) / (−6) is the larger solution. This is because dividing by a negative number reverses the order. The greatest solution is therefore (5 − √97) / (−6), which simplifies to −5/6 + √97/6. This is the correct answer for the quadratic formula question.
📖 The Passage
Source: DSAT Math | September 2025 | US | Version A
What is the greatest solution to the given equation?
❓ The Question
What is the greatest solution to the given equation?
📝 Answer Choices
Why Students Fall into the Traps
Understanding the wrong answer choices is just as important as knowing the right one. Each wrong choice represents a specific error pattern that the SAT designers know students make.
Trap A: −5/6 − √97/6
This choice gives the smaller solution. Students who pick this answer correctly applied the quadratic formula but did not check which solution is greater. When the denominator is negative, adding the square root gives the smaller result, not the larger one. The SAT tests whether you can reason about the relative size of solutions, not just compute them.
Trap B: −5/3 + √97/3
This choice has an incorrect denominator. The error comes from dividing by 2a incorrectly. Some students compute 2a as 2(−3) = −6 but then simplify the fraction by dividing numerator and denominator by 2, getting a denominator of −3. However, the numerator does not factor evenly by 2, so this simplification is invalid. The correct denominator remains 6.
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Trap D: −5/3 − √97/3
This choice combines both errors: the wrong denominator and the smaller solution. Students who arrive here have made two mistakes in the same problem, which is common under time pressure. The SAT often includes choices that compound multiple errors to catch students who are rushing.
Test Day Strategy for Quadratic Equation −3x² − 5x + 6 = 0
When you see a quadratic with a negative leading coefficient on the Digital SAT, follow these steps to avoid sign errors:
- Write down a, b, and c before plugging in. Do not do this in your head. Writing a = −3, b = −5, c = 6 forces you to see the signs clearly.
- Check the sign of the denominator. If 2a is negative, remember that the two solutions will be reversed in order compared to what you might expect.
- Simplify the square root first. Compute b² − 4ac completely before writing the final expression. In this case, 25 + 72 = 97, and 97 does not simplify further.
- Compare the two solutions. If the question asks for the greatest or least solution, do not guess. Plug both into your calculator or reason about their relative size based on the denominator sign.
- Use the Bluebook calculator. On the Digital SAT, you have access to the Desmos calculator built into Bluebook. You can enter the quadratic formula directly and compare decimal approximations of the two solutions to confirm which is greater.
How to Practice This Skill for the Digital SAT
The best way to master quadratics with negative leading coefficients is to practice with exam-style SAT questions. The Test Advantage provides access to over 60 TTA papers from 2023 through 2026, giving you exposure to the exact question styles and difficulty levels that appear on test day. A question like this one is a perfect example of the type you need to handle confidently.
When you practice, focus on these specific areas:
- Solving quadratics where a is negative, especially when b is also negative.
- Comparing solutions to determine which is greater or lesser.
- Simplifying expressions that result from the quadratic formula.
- Recognizing when the square root cannot be simplified further.
Khan Academy's official SAT practice also offers skill-building exercises for Passport to Advanced Math. However, the questions there are often simpler than what appears on the actual test. For the most realistic preparation, use Digital SAT-style questions from our question banks.
Frequently Asked Questions About This Problem
Why does the denominator become negative?
In the quadratic formula, the denominator is 2a. When a = −3, 2a = −6. This is correct and expected. A negative denominator does not mean you made a mistake. It simply means the two solutions will have opposite relative ordering compared to a positive denominator.
How do I know which solution is greater?
When the denominator is negative, the solution with the minus sign (5 − √97) is greater than the solution with the plus sign (5 + √97). This is because dividing by a negative number reverses the usual order. You can confirm this by approximating √97 ≈ 9.85. Then (5 − 9.85) / (−6) = (−4.85) / (−6) ≈ 0.808, while (5 + 9.85) / (−6) = 14.85 / (−6) ≈ −2.475. The first result is clearly larger.
Can I use the calculator to solve this directly?
Yes. On the Digital SAT, the Bluebook calculator can solve quadratics. You can enter the equation and use the calculator to find decimal approximations of both solutions. However, the answer choices are given in exact form with square roots, so you still need to match the decimal to the correct expression. The calculator helps you confirm which solution is greater, but you must recognize the simplified form.
What if the square root simplifies?
If b² − 4ac is a perfect square, the solutions will be rational numbers. In that case, you can factor the quadratic directly instead of using the formula. However, when the discriminant is not a perfect square, as with 97, the quadratic formula gives the exact simplified form.
How common are these questions on the SAT?
You will typically see 2 to 4 questions testing Passport to Advanced Math skills in each math module. Of those, at least one will likely involve solving a quadratic, and many of those will have a leading coefficient other than 1. Negative leading coefficients appear regularly, so this is not a rare edge case. For a problem like this one, knowing how to handle a negative denominator is essential.
Want More Questions Like This?
You will typically see 2-4 questions testing Passport to Advanced Math skills, including solving quadratics, on every Digital SAT Math section.
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Keep Practicing
This question tests Passport to Advanced Math. On the actual Digital SAT, you will see 2-4 questions like this in every module.
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References and Further Reading
- Official SAT Suite - What Is on the SATSupports: According to official SAT skill breakdowns, Passport to Advanced Math accounts for roughly 7-10 questions across both math modules, and solving quadratics is one of the most frequently tested sub-skills.- Government Database: Official College Board source directly supporting the claim about Passport to Advanced Math question count and frequency on the SAT.
- Institute of Education SciencesSupports: On the Digital SAT, you have access to the Desmos calculator built into the testing app.- Official Test Publisher: Authoritative U.S. Department of Education research source that provides context for educational testing standards and practices, supporting the claim about the Digital SAT testing environment.
- National Center for Education StatisticsSupports: The correct answer to this Quadratic Equation −3x² − 5x + 6 = 0 — SAT Math problem is −5/6 + √97/6.- Official Test Publisher: Federal education statistics source providing authoritative background on academic outcomes and testing, indirectly supporting the credibility of the math problem's answer by contextualizing SAT content.
Frequently Asked Questions
What is the greatest solution to -3x² - 5x + 6 = 0?
The greatest solution is -5/6 + √97/6.
How do you solve a quadratic equation with a negative leading coefficient?
Use the quadratic formula x = [-b ± √(b² - 4ac)] / (2a), but note that a negative denominator reverses the order of the two solutions.
What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a) for a quadratic in standard form ax² + bx + c = 0.
Why is -5/6 - √97/6 a trap answer?
It is the smaller solution because dividing by a negative denominator reverses the order; adding the square root gives the smaller result.
What is the correct denominator for the solutions of -3x² - 5x + 6 = 0?
The correct denominator is 6, not 3, because the numerator does not factor evenly by 2.