x + y = S
Factor pairs and quadratic factoring
Find the Number Pair and Factor the Quadratic
Find two numbers that multiply to a product and add to a sum, list every integer pair, explain the signs, solve the related quadratic, or factor ax² + bx + c.Choose the algebra task
Enter coefficients for ax² + bx + c.
Sign and pair logic
Solution
What multiplies to −84 and adds to −8?
The pair is −14 and 6. Their product is −84 because the signs are different, and their sum is −8 because the negative number has the larger magnitude: −14 + 6 = −8.
Another example: what adds to −9 and multiplies to −252?
The integer pair is −21 and 12: their product is −252 and their sum is −9. Enter any product and sum above instead of relying on a page built for only one pair.
Connection to quadratic equations
If x and y have product P and sum S, then they are the roots of t² − St + P = 0. For P = −84 and S = −8, the equation is t² + 8t − 84 = 0, which factors as (t + 14)(t − 6).
Product-and-sum questions
What if no integer pair exists?
The calculator still solves the related quadratic and reports real or complex roots according to the selected domain.
Why do opposite signs give a negative product?
A positive number multiplied by a negative number is negative. The sum takes the sign of the number with larger absolute value.
How does this help factor ax² + bx + c?
The ac method searches for numbers whose product is a·c and whose sum is b, then splits the middle term for grouping.