# Blog Archives

## Quadratic Formula

One of the most useful formulas of all time historically is the solution of the equation:

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

The equation above is known as the quadratic equation.

Theorem: (-b ± √b^{2} – 4ac)/2a is the solution to the quadratic equation.

(1) First, we multiply both sides by 4a and get:

4a^{2}x^{2} + 4abx + 4ac = 0

(2) Next, we add b^{2} – b^{2} to the equation:

4a^{2}x^{2} + 4abx + b^{2} + 4ac – b^{2} = 0

(3) Now, we add b^{2} – 4ac to both sides which gives us:

4a^{2}x^{2} + 4abx + b^{2} = b^{2} – 4ac

(4) Further, we know that:

(2ax + b)^{2} = 4a^{2}x^{2} + 4axb + b^{2}

(5) Combining #4 and #3, gives us:

(2ax + b)^{2} = b^{2} – 4ac

(6) Now, taking the square root of both sides gives us:

2ax + b = ±√b^{2} – 4ac

(7) Now, using basic algebra, we get to:

**x = (-b ±√(b ^{2} – 4ac))/2a.**