# Properties of the roots of quadratic equations - Vieta's formulas

## Properties of the roots of quadratic equations (Vieta's formulas)

### Vieta's formulas:

Sum of the roots:

$X1+X2=- \frac{b}{a}$

Product of the roots:

$X1+X2= \frac{c}{a}$

## Properties of the roots of quadratic equations (Vieta's formulas)

You are probably asking yourselves what Vieta's formulas mean and what this strange name is..
Please, don't be alarmed by the name – Vieta's formulas are very simple and easy formulas that link the roots of a quadratic equation (its solutions) to the coefficients of the different terms.
They are called Vieta after the mathematician who invented them.

$aX^2+bX+c=0$
where $X1$ and $X2$ are the roots of the equation.

Vieta's formulas state that:

The sum of the roots
$X1+X2$ will be equal to: $-\frac{b}{a}$
that is, minus the coefficient of $X$ divided by the coefficient of $X^2$.
$X1+X2= -\frac{b}{a}$

The product of the roots
$X1*X2$ will be equal to: $\frac{c}{a}$
that is, the constant term divided by the coefficient of $X^2$.
$X1+X2= \frac{c}{a}$

We can see that if we divide the entire quadratic equation by $a$, we get:
$X^2+\frac{b}{a}X+\frac{c}{a}=0$

Now, we will use Vieta's formulas and substitute the roots of the equation in place of the relevant coefficients. We get that:

X^2-(X1+X2)b+X1*X2=0

Now, with the help of Vieta's formulas, if we are given the roots of the equation, we can easily construct an equation.
Let's see an example.
If the roots of the equation are given –
$X1=3$

$X2=4$
We can find out what the quadratic equation is according to the formulas.

$X^2-(3+4)x+(3*4)=0$
We get:
$X^2-7x+12=0$

(Image 1: The product of the roots is the constant term, minus the sum of the roots equals the coefficient of b)

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