For ax² + bx + c = 0 with roots α and β, the sum of the roots is −b/a and the product of the roots is c/a. These let you answer questions about the roots without solving the equation.
This lesson is part of quadratic functions. Next to it, using the discriminant to classify roots tells you how many roots there are.
Where do the formulas come from?
If α and β are the roots, then x² + (b/a)x + c/a = (x − α)(x − β) = x² − (α + β)x + αβ. Comparing the coefficients gives α + β = −b/a and αβ = c/a.
Remember where they come from, because that lets you rebuild the formulas if you forget a sign.
Worked example: sum and product
Here is an original example. The equation 2x² − 5x + 3 = 0 has roots α and β. Find α + β and αβ.
Here a = 2, b = −5, c = 3.
α + β = −b/a = −(−5)/2 = 5/2.
αβ = c/a = 3/2.
Check by solving: 2x² − 5x + 3 = (2x − 3)(x − 1), so the roots are 3/2 and 1. Their sum is 5/2 and product is 3/2. Both match.
Finding an unknown from a relationship between roots
Suppose one root of x² − 9x + k = 0 is twice the other. Find k.
- Let the roots be α and 2α.
- Sum: α + 2α = 9, so 3α = 9 and α = 3.
- The roots are 3 and 6.
- Product: k = 3 × 6 = 18.
Check: x² − 9x + 18 = (x − 3)(x − 6). The roots are 3 and 6, and 6 is twice 3.
Forming a new equation
Sometimes the question asks for an equation whose roots are related to another equation’s roots. Let α and β be the roots of x² − 7x + 10 = 0. Form the equation with roots α + 1 and β + 1.
- From the original: α + β = 7 and αβ = 10.
- New sum: (α + 1) + (β + 1) = 7 + 2 = 9.
- New product: (α + 1)(β + 1) = αβ + (α + β) + 1 = 10 + 7 + 1 = 18.
- Equation: x² − 9x + 18 = 0.
Check by solving the original. The roots are 2 and 5, so the new roots are 3 and 6. Their sum is 9 and product is 18. It matches.
The mistake that costs marks
The common slip is to forget the minus sign in front of the sum. Some students write x² + 9x + 18 = 0, which has roots −3 and −6, not 3 and 6.
| Step | Wrong | Right |
|---|---|---|
| Form | x² + (sum)x + (product) | x² − (sum)x + (product) |
| Equation | x² + 9x + 18 = 0 | x² − 9x + 18 = 0 |
| Roots | −3 and −6 | 3 and 6 |
The check is quick. Substitute one intended root, such as x = 3. In the wrong equation, 9 + 27 + 18 is not zero. In the right one, 9 − 27 + 18 is zero.
Check yourself
The roots of 3x² − 12x + 7 = 0 are α and β. Find α + β, αβ, and the equation with roots 2α and 2β.
Answer
Here a = 3, b = −12, c = 7.
α + β = 12/3 = 4, and αβ = 7/3.
For roots 2α and 2β: sum = 2(α + β) = 8, and product = 4αβ = 28/3.
Equation: x² − 8x + 28/3 = 0, or 3x² − 24x + 28 = 0.
What to study next
Knowing the roots together is one tool. The next is knowing how many roots there are without solving. Continue with using the discriminant to classify roots, then try the practice set.
If you want a teacher to spot when sum and product is the quick route, see online one-to-one Additional Mathematics tuition.