math
10th Grade - Quadratic equations

10th Grade lesson

1 - Quadratic equations


How to solve the equation

Here are some examples of quadratic equations :

quadratic equation, quadratic equation, and quadratic equation.

These equations are of the form general form quadratic equation. 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 math symbol means "is equivalent to" and that symbole delta is a greek letter called "delta".


You have to solve this kind of equation general quadratic 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.

canonical form

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

A product of factors is null if and only if one of the factors is null. So :
quadratic equation solutions
quadratic equation solutions
or

or
quadratic equation solutions
quadratic equation solutions



To sum up, if you want to solve an equation like equation du second degre, for example quadratic equation example, first calculate the number calcul de delta, and then, there are three cases :
If delta negatif, then the equation has no solution,
If delta equals zero, then the equation has only one solution which is delta equals zero solution unique, and
If delta positif, then the equation has two solutions, which in this case are :
quadratic equation solutions
quadratic equation solutions
To train, you can check that the equation quadratic equation has no solution, that the equation quadratic equation has only one solution which is x = - 1, and that the equation quadratic equation has two solutions which are quadratic equation and x = -2,5.

Graph

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

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 quadratic equation. For example,
If a > 0 and delta, the vertex of the parabola is at the bottom and the parabola intersects twice with the X-axis :
math
If a < 0 and delta , the vertex is at the top and the parabola never intersects with the X-axis :
math

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

Inequalities

To solve this kind of equations sign of quadratic function, you start by solving the equation quadratic 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 :
quadratic equation solutions



See also : algebra on fmaths.com



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Quadratic equation

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