To solve a quadratic equation, ax² + bx + c = 0, we must find values of the unknown variable x which make the left-hand and right-hand sides equal. Such values are known as solutions or roots of the quadratic equation.

Factored quadratic equation can be solved using the zero-product principle.

Zero-Product principle

If the product two numbers is zero, then one of these factors must be zero. That is, for any real numbers p and q, if p × q = 0, then either p = 0 or q = 0.

To solve the quadratic equation ax² + bx + c = 0, we factorize the expression ax² + bx + c into linear factors, whenever possible, and solve the linear equation using the fact that p × q = 0.

1. Solving Quadratic Equations of the form ax² + bx = 0 by Factorization

The quadratic expression of the form ax² + bx = 0 can be factorized as:
ax² + bx = 0
x (ax + b) = 0
x = 0 or ax + b = 0

2. Solving Quadratic Equations of the form a²x² - b² = 0 by Factorization.

Using the difference of two squares formula: a² - b² = (a + b) (a - b), the quadratic equation of the form x² - k² = 0 can be factorized as (x + k)(x - k) = 0 which implies that x = -k or x = k.