Sum of the roots:

Sum of the roots:

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

Product of the roots:

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

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.

Let's recall the quadratic equation that we are already familiar with:

$aX^2+bX+c=0$

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)

Related Subjects

- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- Ways to represent a quadratic function
- Various Forms of the Quadratic Function
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- The quadratic function
- Quadratic Inequality
- Parabola
- Symmetry in a parabola
- Plotting the Quadratic Function Using Parameters a, b and c
- Finding the Zeros of a Parabola
- Methods for solving a quadratic function
- Completing the square in a quadratic equation
- Squared Trinomial
- Families of Parabolas
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)