Sum and product of roots: S = −b/a, P = c/a

Vieta’s formulas

For a quadratic $a x^2 + b x + c = 0$ with two roots $x_1$ and $x_2$:

$S = x_1 + x_2 = -\dfrac{b}{a}$

$P = x_1 \times x_2 = \dfrac{c}{a}$

Why is it useful?

• Quickly verify a factorization.
• Find the second root if you know one.
• Build a polynomial with given roots.
• Study the sign of the roots without computing them.

Find the 2nd root

Example: $x^2 - 7x + 12 = 0$. We see $x = 3$ is a root (since $9 - 21 + 12 = 0$).

Sum: $x_1 + x_2 = 7$, so $x_2 = 7 - 3 = 4$.

Product check: $3 \times 4 = 12$. ✓

Build a polynomial

Find a polynomial with roots $2$ and $-5$.

$S = 2 + (-5) = -3$, $P = 2 \times (-5) = -10$.

$P(x) = x^2 - S x + P = x^2 + 3x - 10$.

Sign of roots (without computation)

• Product $P > 0$: both roots have the same sign.
• Product $P < 0$: they have opposite signs.
• Sum $S > 0$ and $P > 0$: two positive roots.
• Sum $S < 0$ and $P > 0$: two negative roots.

Careful

These formulas only work when the quadratic has two roots (real or complex), i.e. $\Delta \neq 0$ or $= 0$ (double root: $S = 2x_0$).

If $\Delta < 0$ in $\mathbb{R}$: formulas hold in $\mathbb{C}$ (roots are conjugate).

Common mistakes

1. Forgetting the $-$ in $S = -b/a$.
2. Writing $S = -b$ and $P = c$ and forgetting the factor $a$.
3. Using formulas for a non-normalized quadratic without dividing by $a$.

?À toi de jouer

x² − 9x + 20 = 0. Sum of roots?

A.9B.−9C.20D.−20

Valider