### Quadratic Equation

The quadratic formula is a mathematical formula used to solve quadratic equations. It is derived using the coefficients of the terms in the equation and allows for the calculation of the roots of the equation. The formula is typically written

`\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]`

where a, b, and c are the coefficients of the quadratic equation. This formula enables you to find the roots of the quadratic equation, and it is derived by completing the square.

### Other quadratic equation formulae and theorems

Another important formula is the discriminant formula which is used to determine the nature of the roots of a quadratic equation. The discriminant is given by

#### Discriminant of quadratic equations

The discriminant is the expression found under the square root sign in the quadratic formula. It determines the nature of the roots of a quadratic equation. For a quadratic equation, the discriminant is given by the formula:

`\[Δ=b^2-4ac\]`

There are three possible cases depending on the value of the discriminant:

`\[1.\ \ If\ \ Δ>0\]`

The quadratic equation will have two distinct real roots. This means the graph of the quadratic equation will intersect the x-axis at two different points.

`\[2.\ \ If\ \ Δ=0\]`

The quadratic equation will have one real root with a multiplicity of 2. This means the graph of the quadratic equation will touch the x-axis at one point.

`\[3.\ \ If\ \ Δ<0\]`

The quadratic equation will have two complex conjugate roots. This means the graph of the quadratic equation will not intersect the x-axis.

In summary, the conditions for the discriminant of a quadratic equation are:

1) If Δ > 0, the equation has two distinct real roots.

2) If Δ = 0, the equation has one real root with a multiplicity of 2.

3) If Δ < 0, the equation has two complex conjugate roots.