# Quadratic Equations

Linear Equation can have degree of at most 1 and has only one solution. **Quadratic Equation** can have degree of at most 2 and has two solutions.

General form of quadratic equation: ax^{2} + bx + c = 0, where a, b and c are constants. Note that maximum degree of x is 2.

### Solving Quadratic Equations

Unlike linear equations, any quadratic equation always has two solutions called roots of quadratic equation. After solving quadratic equation, you will get two values of x. To solve quadratic equation, you can use directly quadratic formula.

x = [-b ± √(b^{2}-4ac)]/2a

### Discriminant

In the above quadratic formula, the expression underneath the square root sign is called the **discriminant** of the quadratic equation. Discriminant is used to find the nature of roots.

Δ = b^{2} - 4ac

Case 1: Δ > 0

Real and distinct roots

Case 2: Δ = 0

Real and one distinct root (two same roots)

Case 3: Δ < 0

Roots are imaginary and occur as complex conjugates of each other

### Sum and Product of Roots

Let α and β be the roots of quadratic equation x^{2 }+ px + q = 0

x^{2 }+ px + q = (x-α)(x-β)

x^{2 }+ px + q = x^{2 }- (α+β)x + αβ

p= -(α+β)

q = αβ

Sum of the roots} = -p

Product of roots} = q