To find the solution of a quadratic equation ax^{2 }+ bx + c = 0 we use the quadratic formula, which has the following form:

**x = [-b ± √ (b ^{2 }- 4ac)] / 2a**

Substituting the values of the coefficients a, b and c into it we can easily obtain the two roots of x, which satisfy this equation.

Part "b^{2 }- 4ac = D ”is called **discriminating **and depending on its value, we will know the **nature of the roots** from the quadratic equation:

- If D> 0, the roots of that equation will be real and different.
- If D = 0, the roots of the equation will be real and equal.
- If D <0, the roots will be complex.

## Examples:

**1. Find the nature of the roots of the following equation: ****20x² - x - 1 = 0.**

The coefficients are: a = 20; b = - 1 and c = - 1. By replacing them in the discriminant D = b^{2 }- 4ac, we have:

D = (- 1)^{2 }- 4 (20) (- 1) = 1 + 80 = 81.

Since D> 0, the roots will be **real and different**.

**2. Find the nature of the roots of the following equation: ****4x² - 16x + 16 = 0.**

The coefficients are: a = 4; b = - 16 and c = 16. When replacing them in the discriminant D = b^{2 }- 4ac, we have:

D = (- 16)^{2 }- 4 (4) (16) = 256 - 256 = 0.

Since D = 0, the roots will be **real and equal**.

**3. Find the nature of the roots of the following equation: ****4x² - 8x + 7 = 0.**

The coefficients are: a = 4; b = - 8 and c = 7. By replacing them in the discriminant D = b^{2 }- 4ac, we have:

D = (- 8)^{2 }- 4 (4) (7) = 64 - 112 = - 48.

Since D <0, the roots will be **complex**.