One of the most useful formulas of all time historically is the solution of the equation:
ax2 + bx + c = 0.
The equation above is known as the quadratic equation.
Theorem: (-b ± √b2 - 4ac)/2a is the solution to the quadratic equation.
(1) First, we multiply both sides by 4a and get:
4a2x2 + 4abx + 4ac = 0
(2) Next, we add b2 - b2 to the equation:
4a2x2 + 4abx + b2 + 4ac - b2 = 0
(3) Now, we add b2 - 4ac to both sides which gives us:
4a2x2 + 4abx + b2 = b2 - 4ac
(4) Further, we know that:
(2ax + b)2 = 4a2x2 + 4axb + b2
(5) Combining #4 and #3, gives us:
(2ax + b)2 = b2 - 4ac
(6) Now, taking the square root of both sides gives us:
2ax + b = ±√b2 - 4ac
(7) Now, using basic algebra, we get to:
x = (-b ±√b2 - 4ac)/2a.