SAT Product of Roots: Formula and Worked Examples

For a quadratic ax² + bx + c = 0, the product of the two roots is c/a. The leading coefficient matters: divide the constant term by the coefficient of x². Do not use -b/a, which is the sum of the roots.

The three examples below are real SAT-style questions from our SAT Math question banks. We solved each one by hand and checked it against the answer key. Being in our bank does not mean College Board wrote the question or that it came from a specific test date.

Why the product is c/a

Write the quadratic as a(x - r₁)(x - r₂). Expanding gives ax² - a(r₁ + r₂)x + ar₁r₂. Comparing constant terms with ax² + bx + c shows ar₁r₂ = c, so r₁r₂ = c/a.

College Board includes quadratic and polynomial equations in Advanced Math. The formula is useful, but the question may instead give a factored polynomial or ask about real solutions. Read the exact wording.

Example 1: direct use of c/a

4x² − 4x − 24 = 0

What is the product of the solutions to the given equation?

From our June 2026 SAT Math question bank. Choices: -24, -6, 6, 24.

Here a = 4 and c = -24. Therefore the product is c/a = -24/4 = -6, matching the answer key.

Notice that b = -4 is irrelevant to the product. Using -b/a would answer the sum question instead.

Example 2: solve for a coefficient multiplier

21x² + (21s + 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?

From our December 2024 SAT Math question bank. Choices: 1/21, 21, 1, 1/24.

The leading coefficient is 21 and the constant term is rs. The product of the roots is:

c/a = rs/21

The question calls this product krs. Because r and s are positive, rs is nonzero and can be divided out:

rs/21 = krs

So k = 1/21, matching the answer key. The middle coefficient, 21s + r, is a distractor for this particular question.

Example 3: multiply roots from every factor

$\frac{5}{9}$(5x + 9)(x + √(5k + 9))(x - √(5k + 9)) = 0

In the given equation, k is a positive constant. The product of the solutions to the equation is 72. What is the value of k?

From our May 2025 SAT Math question bank. Choices: 4, 5.5, 6.2, 7.

The nonzero factor 5/9 does not create a solution. The three roots are:

• 5x + 9 = 0, so x = -9/5.
• x + √(5k + 9) = 0, so x = -√(5k + 9).
• x - √(5k + 9) = 0, so x = √(5k + 9).

The two square-root factors multiply to -(5k + 9). Multiplying all three roots gives:

(-9/5)[-(5k + 9)] = 9(5k + 9)/5

Set this equal to 72. Then 9(5k + 9)/5 = 72, so 5k + 9 = 40 and k = 31/5 = 6.2, matching the answer key.

Real roots and the discriminant

The identity c/a applies to the two roots counted with multiplicity, including complex roots. But if a problem says “real solutions,” check the discriminant b² - 4ac. A negative discriminant means the quadratic has no real roots even though its two complex roots still have product c/a.

This is why a coefficient calculation alone cannot prove that two real x-intercepts exist. Graphing can help, but the discriminant gives an exact check.

For the graphing check, our complete guide to using Desmos on the SAT shows how to open the same calculator version the test uses.

Common mistakes

• Using the wrong formula: sum = -b/a; product = c/a.
• Dropping the denominator: a is the full coefficient of x², including a fraction.
• Treating a global nonzero factor as a root: it changes scale, not zeros.
• Ignoring multiplicity or domain: read whether the question asks for roots, distinct roots or real solutions.
• Inventing missing options: solve from the full question and compare only with the choices given.

For every formula you need on test day, see our list of SAT math formulas. For the companion coefficient relationship, read the sum-of-solutions guide. To graph and check intersections, use the systems guide. For fitting data rather than finding zeros, use the regression guide.

Continue practicing

Start with the free quadratic lesson notes and write “sum or product?” before using a coefficient formula. For broader access, compare current TTA plans and included access. Check the current terms before subscribing.