Quadratic Equations • Topic 4 of 4

Sum & Product of Roots

For ax² + bx + c = 0 with roots p and q, the roots satisfy p + q = −b/a and p × q = c/a. This lets you answer many GRE questions without ever solving the equation — if a problem asks for the sum or product of the solutions, read it straight off the coefficients. For a monic quadratic x² + bx + c = 0 it is even cleaner: the sum of the roots is −b and the product is c. The relationship also runs backwards: to build a quadratic with given roots, use x² − (sum)x + (product) = 0, so roots 3 and −5 give x² + 2x − 15 = 0. Use it as a sanity check after factoring too — the two numbers you found should add to −b/a and multiply to c/a. The single most useful GRE habit here is to stop and ask "does the question actually need the individual roots?" Often it does not, and Vieta's relations turn a full solve into one subtraction.

✅ Solved examples

1. The equation x² − 7x + 10 = 0 has roots p and q. Find p + q and pq without solving.
For a monic quadratic, sum = −b = 7 and product = c = 10. (Check: roots are 2 and 5.)
2. If the roots of 2x² − 8x + 6 = 0 are p and q, find p + q.
Sum = −b/a = 8/2 = 4. (You never need the actual roots 1 and 3.)
3. Build a quadratic (leading coefficient 1) whose roots are 4 and −6.
Sum = 4 + (−6) = −2; product = 4 × (−6) = −24. Equation: x² − (−2)x + (−24) = x² + 2x − 24 = 0.
4. One root of x² + kx + 12 = 0 is 3. Find the other root and k.
Product of roots = 12, so the other root is 12/3 = 4. Sum of roots = 3 + 4 = 7 = −k, so k = −7.

✏️ Practice — try these, take hints as needed

1. For x² − 11x + 24 = 0, find the sum and product of the roots without solving.
Monic quadratic.
Sum = −b, product = c.
b = −11, c = 24.
Sum = 11, product = 24
2. The roots of 3x² + 12x − 15 = 0 are p and q. Find p + q.
Sum = −b/a.
−12/3.
Simplify.
p + q = −4
3. Find a monic quadratic whose roots are 5 and 2.
Sum = 7, product = 10.
x² − (sum)x + product.
Fill in.
x² − 7x + 10 = 0
4. One root of x² − 9x + c = 0 is 4. Find the other root.
Sum of roots = 9.
Other root = 9 − 4.
Then c = product if needed.
Other root = 5
5. The roots of x² + bx + 20 = 0 are 2 and 10. Find b.
Sum of roots = −b.
2 + 10 = 12.
−b = 12.
b = −12

📝 Topic test — 8 questions

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