Category Archives: Quadratic equation

Quadratic Formula


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.

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