# Solutions to Quadratic Equations

### ax^{2} + bx +c = 0

**b ^{2} − 4ac ** is called the

**discriminant**of the quadratic equation and determines the number of solutions in each equation. Three cases can be distinguished:

### b^{2} − 4ac > 0

In this case, the quadratic equation has **two solutions**, which are distinct real numbers.

### b^{2} − 4ac = 0

In this case, the quadratic equation has a **double solution**.

### b^{2} − 4ac < 0

In this case, the quadratic equation has no real solutions.

## Properties of the Solutions to Quadratic Equations

The sum of the solutions of a quadratic equation is:

The product of the solutions of a quadratic equation is:

### Quadratic Equation from Solutions

If the roots of an equation are known, it can be writen as:

Where:

**S = x _{1 }+ x_{2 } **

** P = x _{1 } · x_{2 } **

Write a quadratic equation whose solutions are: 3 and −2.

S = 3 − 2 = 1

P = 3 · (−2) = −6

x^{2} − x − 6 = 0

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