Formula

Sum and product of roots (Vieta)

Algebra • Quadratic Equations • Roots

Formula

α + β = - \frac{b}{a}, α β = \frac{c}{a}

α, β

are the roots of

a x^{2} + b x + c = 0

, then

α + β = - \frac{b}{a}

and

α β = \frac{c}{a}

How to use it

1Answers symmetric questions about roots without solving the equation.
2To build a quadratic from a known sum $s$ and product $p$ : $x^{2} - s x + p = 0$ .
3Combine with the square-of-sum identity to reach $α^{2} + β^{2}$ or $∣ α - β ∣$ .

Find

α^{2} + β^{2}

for

x^{2} - 7 x + 12 = 0

Step-by-step

Answer

25

Watch out

The signs trip people up: the sum is

- b / a

, not

b / a

Builds on

Also known as

vieta's formulas, sum of roots, product of roots