Vieta’s Formulas: Sum and Product of Roots (with Examples)

Vieta’s formulas are a favorite shortcut for handling quadratic equations quickly. This guide covers the direct form (the sum and product of the roots), the converse (finding two numbers from their sum and product), plus the classic tricks for reading roots by inspection and determining the signs of the roots.

The sum and product of the roots

Take a quadratic $a{x}^2+bx+c=0$ with $a\ne 0$. If it has two roots $x_1$ and $x_2$ (that is, the discriminant $\Delta =b^2-4ac\ge 0$), then:

$$S=x_1+x_2=-\frac{b}{a}P=x_1\cdot x_2=\frac{c}{a}$$

The powerful part is that you know the sum and product of the roots without solving the equation. Keep the condition $\Delta \ge 0$ in mind: if $\Delta <0$ the equation has no real roots, so there is no real sum or product to apply the formulas to.

For example, take $x^2-5x+6=0$ (here $a=1$, $b=-5$, $c=6$):

$$S=-\frac{-5}{1}=5P=\frac{6}{1}=6$$

So the two roots add up to $5$ and multiply to $6$.

Finding roots by inspection

Once you know $S$ and $P$, you just need two numbers whose sum is $S$ and whose product is $P$. In the example above, the two numbers with sum $5$ and product $6$ are $2$ and $3$. So the roots are $x_1=2$ and $x_2=3$ — no discriminant needed.

Two lightning-fast special cases

There are two special cases where you can read the roots off directly, just by adding up the coefficients:

For instance, $2x^2-5x+3=0$ has $a+b+c=2-5+3=0$, so immediately $x_1=1$ and $x_2=\frac{3}{2}$.

And $3x^2+8x+5=0$ has $a-b+c=3-8+5=0$, so $x_1=-1$ and $x_2=-\frac{5}{3}$.

Determining the signs of the roots

Vieta’s formulas let you decide the signs of the two roots without computing them. First you always need $\Delta \ge 0$ so that real roots exist, then you look at $S$ and $P$:

Note: when $P<0$ you automatically have $\Delta >0$ (a negative product forces the roots to be distinct), so you don’t need to check $\Delta$ separately. When $P>0$, however, you must confirm $\Delta \ge 0$ first before using $S$ to read off the signs.

The converse: finding two numbers from their sum and product

This is the reverse direction, and it comes up constantly. If two numbers have sum $S$ and product $P$, then they are the roots of:

$$X^2-SX+P=0$$

Such numbers exist (that is, this equation has real roots) if and only if $S^2-4P\ge 0$.

Example. Find two numbers whose sum is $10$ and whose product is $21$.

The numbers we want are the roots of:

$$X^2-10X+21=0$$

Compute the discriminant: $\Delta =(-10{)}^2-4\cdot 21=100-84=16>0$. So:

$$X_{1,2}=\frac{10\pm \sqrt{16}}{2}=\frac{10\pm 4}{2}\Rightarrow X_1=7,X_2=3$$

The two numbers are $7$ and $3$ (check: $7+3=10$ and $7\cdot 3=21$, correct).

Application: symmetric expressions in the roots

A very common problem type asks you to evaluate an expression involving both roots without solving the equation.

Example. Given $x^2-7x+5=0$ with roots $x_1$ and $x_2$, find $x_1^2+x_2^2$.

By Vieta’s formulas: $S=x_1+x_2=7$ and $P=x_1{x}_2=5$. Rewrite the target expression in terms of $S$ and $P$:

$$x_1^2+x_2^2=(x_1+x_2{)}^2-2x_1{x}_2=S^2-2P$$

Substitute the values:

$$x_1^2+x_2^2=7^2-2\cdot 5=49-10=39$$

So $x_1^2+x_2^2=39$. This is much faster and cleaner than finding each root and squaring it.

Common mistakes

• Forgetting to check $\Delta \ge 0$. You may only use Vieta’s formulas when the equation has real roots. Many trick problems hand you a quadratic with no real solutions.
• Getting the sign of $S$ wrong. The formula is $S=-\frac{b}{a}$ — don’t drop the minus sign. If $b=-5$ then $S=5$, not $-5$.
• Forgetting to divide by $a$. When $a\ne 1$, you must use $S=-\frac{b}{a}$ and $P=\frac{c}{a}$, not just $-b$ and $c$.
• Misremembering the converse condition. Two numbers with sum $S$ and product $P$ exist only when $S^2-4P\ge 0$.

Master the two formulas $S=-\frac{b}{a}$ and $P=\frac{c}{a}$ along with the converse, and you’ll read off roots and evaluate symmetric expressions far more quickly. Try entering the examples above into the calculator to check each step of your own work.