Mathwords logoMathwords

Vieta's Formulas — Definition, Formula & Examples

Vieta's Formulas are a set of equations that express the sum, product, and other symmetric combinations of a polynomial's roots directly in terms of its coefficients, without requiring you to solve for the roots themselves.

For a polynomial anxn+an−1xn−1+⋯+a1x+a0=0a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0 with roots r1,r2,…,rnr_1, r_2, \ldots, r_n, Vieta's Formulas state that the kk-th elementary symmetric polynomial of the roots equals (−1)kan−kan(-1)^k \frac{a_{n-k}}{a_n}. In particular, r1+r2+⋯+rn=−an−1anr_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n} and r1⋅r2⋯rn=(−1)na0anr_1 \cdot r_2 \cdots r_n = (-1)^n \frac{a_0}{a_n}.

Key Formula

r1+r2=−ba,r1⋅r2=car_1 + r_2 = -\frac{b}{a}, \quad r_1 \cdot r_2 = \frac{c}{a}
Where:
  • a,b,ca, b, c = Coefficients of the quadratic $ax^2 + bx + c = 0$
  • r1,r2r_1, r_2 = The two roots of the quadratic equation

How It Works

For a quadratic ax2+bx+c=0ax^2 + bx + c = 0 with roots r1r_1 and r2r_2, the formulas give you two relationships: r1+r2=−bar_1 + r_2 = -\frac{b}{a} and r1⋅r2=car_1 \cdot r_2 = \frac{c}{a}. For a cubic ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0 with roots r1,r2,r3r_1, r_2, r_3, you get three: the sum of roots is −ba-\frac{b}{a}, the sum of products taken two at a time is ca\frac{c}{a}, and the product of all three roots is −da-\frac{d}{a}. The pattern continues for any degree. These formulas are especially powerful in competition math, where you might be asked about sums or products of roots without ever finding the roots individually.

Worked Example

Problem: The equation 2x2−10x+8=02x^2 - 10x + 8 = 0 has roots r1r_1 and r2r_2. Find r1+r2r_1 + r_2 and r1⋅r2r_1 \cdot r_2 without solving the equation.
Identify coefficients: Here a=2a = 2, b=−10b = -10, and c=8c = 8.
Apply sum formula: The sum of the roots equals −b/a-b/a.
r1+r2=−−102=5r_1 + r_2 = -\frac{-10}{2} = 5
Apply product formula: The product of the roots equals c/ac/a.
r1⋅r2=82=4r_1 \cdot r_2 = \frac{8}{2} = 4
Answer: The sum of the roots is 55 and the product of the roots is 44. (You can verify: the roots are 11 and 44, which indeed sum to 55 and multiply to 44.)

Why It Matters

Vieta's Formulas appear constantly in math competitions (AMC, AIME, olympiads) where problems ask about root relationships without expecting you to factor or use the quadratic formula. They also underpin techniques in precalculus and linear algebra, such as connecting the trace and determinant of a matrix to its eigenvalues.

Common Mistakes

Mistake: Forgetting the alternating signs in the general formula and writing the sum of roots as +b/a+b/a instead of −b/a-b/a.
Correction: Remember the sign pattern: the sum of roots always carries a factor of (−1)1=−1(-1)^1 = -1. For ax2+bx+c=0ax^2 + bx + c = 0, the sum is −b/a-b/a, not b/ab/a.