AlgebraAMC 10

Vieta's Formulas (Quadratic)

r_1 + r_2 = -\frac{b}{a}, \qquad r_1 r_2 = \frac{c}{a}

For $ax^2 + bx + c = 0$ with roots $r_1, r_2$ .

What it says

Vieta's formulas tie the roots of a quadratic directly to its coefficients. If you only need the sum or product of the roots, or anything built from them like $r^2 + s^2 = (r+s)^2 - 2rs$ , you never have to solve the equation. This is one of the most-used shortcuts on the AMC.

Show proof

If the roots are $r_1, r_2$ , the quadratic factors as $a(x - r_1)(x - r_2) = ax^2 - a(r_1 + r_2)x + ar_1r_2$ . Matching coefficients with $ax^2 + bx + c$ gives $-a(r_1 + r_2) = b$ and $ar_1r_2 = c$ .

When to use it

The problem asks about the sum, product, or a symmetric expression of the roots without needing the roots themselves.

Example

The roots of $2x^2 - 7x + 3 = 0$ are $p$ and $q$ . Find $p^2 + q^2$ .

Show solution

$p+q = \frac72$ and $pq = \frac32$ , so $p^2+q^2 = (p+q)^2 - 2pq = \frac{49}{4} - 3 = \frac{37}{4}$ . Answer: $\frac{37}{4}$