Viète’s formulas: sums, products, and real-root conditions
Use root sums and products and their converse while retaining the existence, multiplicity, and distinctness conditions for real roots.
Content updated: October 22, 2026
Viète’s formulas relate the coefficients of a polynomial to the sum and product of its roots. They can avoid solving explicitly for each root, but they do not replace the conditions for real roots or for distinct roots.
Learning goals: use root sums and products, construct an equation from two real numbers, and check the necessary conditions before using a parameter.
Prerequisites: real quadratic equations, discriminants, and polynomial factorization.
On this page: Definition · Method · Examples · Practice · Further learning
1. The forward formulas
Let $f(x)=ax^2+bx+c$, where $a\ne0$. If the equation has two real roots $x_1,x_2$, allowing equality when roots are counted with multiplicity, then
Indeed, $f(x)=a(x-x_1)(x-x_2)$. Expanding and comparing the coefficient of $x$ and the constant term proves both identities. A requirement of distinct roots additionally needs $\Delta>0$; real roots alone require $\Delta\ge0$.
Writing $S=-b/a,P=c/a$ alone does not prove the existence of two real roots. For example, $x^2+1=0$ gives the formal values $S=0,P=1$, but no two real numbers have this sum and product because $S^2-4P=-4<0$.
2. The converse and recognizing simple roots
For given real numbers $u,v$, set $S=u+v$ and $P=uv$. Then
Thus $u,v$ are roots of the equation obtained by setting the left side equal to $0$. Conversely, real numbers with sum $S$ and product $P$ exist exactly when $S^2-4P\ge0$; they are distinct exactly when this inequality is strict. If the discriminant equals $0$, both numbers equal $S/2$.
Two useful shortcuts also require $a\ne0$. If $a+b+c=0$, then $f(1)=0$ and the roots counted with multiplicity are $1,c/a$. If $a-b+c=0$, then $f(-1)=0$ and the roots counted with multiplicity are $-1,-c/a$. Either pair may coincide, so distinctness must not be assumed.
For symmetric expressions, one may use $x_1^2+x_2^2=S^2-2P$. With a parameter, first impose the root-existence conditions, simplify using $S,P$, and check the resulting parameter values against those conditions.
3. Worked examples
Example 1. Recognizing roots from their sum and product
Consider $x^2-5x+6=0$. The numbers $2,3$ have sum $5$ and product $6$, so the converse shows that they are roots. The factorization $(x-2)(x-3)=0$ also confirms two distinct real roots.
Example 2. Coefficients summing to zero
For $x^2-3x+2=0$, we have $a+b+c=1-3+2=0$. One root is $1$, and the other is $c/a=2$. Since $1\ne2$, the roots are distinct.
Example 3. Evaluating at −1
For $x^2+5x+4=0$, we have $a-b+c=1-5+4=0$. One root is $-1$, and the other is $-c/a=-4$. The factorization $(x+1)(x+4)=0$ checks them directly.
Example 4. A symmetric condition on two roots
Find $m$ such that $x^2-2mx+4=0$ has two distinct roots satisfying $x_1^2+x_2^2=x_1x_2+24$. First require $\Delta'=m^2-4>0$, that is, $m<-2$ or $m>2$.
By Viète’s formulas, $S=2m$ and $P=4$. The given equality is equivalent to $S^2-2P=P+24$, or $4m^2-36=0$. Thus $m=3$ or $m=-3$. Both satisfy the distinctness condition and are accepted. Explicit radical expressions for the roots are unnecessary for checking this symmetric equality.
4. Practice
- Find two real numbers with sum $7$ and product $12$.
- Find $m$ such that $x^2-2x+m=0$ has two distinct positive real roots.
- Are two numbers with sum $-4$ and product $4$ distinct? Find them.
Hints and answers
- The two numbers are roots of
$$X^2-7X+12=(X-3)(X-4)=0.$$
Thus the numbers are $3,4$.
- The distinctness condition is $m<1$. Then $S=2>0$ and $P=m>0$ ensure that both roots are positive: a positive product gives the same sign, and a positive sum excludes both being negative. Therefore $0<m<1$.
- The sum–product discriminant is $(-4)^2-4\cdot4=0$, so both numbers equal $-2$ and are not distinct.
5. Common mistakes
Do not use Viète’s formulas to bypass real-root conditions. Do not change the root $-1$ to $1$ when $a-b+c=0$. Distinguish two roots counted with multiplicity from two different values. If the coefficient $a$ may equal $0$, consider the degenerate equation before dividing.
6. Further learning
Quadratic equations (English lesson) · The sign of a quadratic polynomial (English lesson) · Comparing quadratic roots with a number $\alpha$ (English lesson).
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