## How to solve the equation

Here are some examples of quadratic equations :

, , and .

These equations are of the form . It's long to solve this kind of equations, so if you want, you can skip the part between the two green lines. But you have to know that means "is equivalent to" and that is a greek letter called "delta".

You have to solve this kind of equation . a is different from 0 because on the contrary it would be a first degree equation. Divide the whole equation by a, then you just have to do some calculations and factorizations.

To simplify the calculations, on the last line, we supposed that . We notice that :
If then the equation can't have a solution (a positive number minus a negative number can't be equal to zero).
If , then :
If :

A product of factors is null if and only if one of the factors is null. So :
 oror

To sum up, if you want to solve an equation like , for example , first calculate the number , and then, there are three cases :
If , then the equation has no solution,
If , then the equation has only one solution which is , and
If , then the equation has two solutions, which in this case are :

To train, you can check that the equation has no solution, that the equation has only one solution which is x = - 1, and that the equation has two solutions which are and x = -2,5.

## Graph

The graph of such a function is a parabola. The coordinates of the S summit of the parabola are :

If a is positive, then the vertex is at the bottom of the parabola, on the contrary it is at the top the parabola. The parabola intersects the X-axis as many times as the numbers of solutions of the equation . For example,
 If a > 0 and , the vertex of the parabola is at the bottom and the parabola intersects twice with the X-axis :
 If a < 0 and , the vertex is at the top and the parabola never intersects with the X-axis :

These considerations are useful when you want to solve a quadratic equation.

## Inequalities

To solve this kind of equations , you start by solving the equation , then you do a drawing such as the one above to determine the shape of the curve, then you read the solutions on the graph. In that case :