Proof of the quadratic formula

The following is a proof of the quadratic formula. It will show you how the quadratic formula, which is widely used has been developed.

The evidence is performed using the standard form of a quadratic equation and the standard form of problems by completing the square

Start with the form of a standard quadratic equation:

ax2 + bx + c = 0

Divide both sides of the equation by a kind you can fill the place

quadratic-formula-proof-image

quadratic-formula-proof-image

Subtract c/a leave both sides

quadratic-formula-proof-image

quadratic-formula-proof-image

Complete the square:

The coefficient of the second term is b / a

This coefficient divided by 2 and square the result to get (a b/2) 2

Add 2 (b/a) 2 on both sides:

quadratic-formula-proof-image

Since the left side of the right side of the equation above is a perfect square, you can factor the left-hand side by using the coefficient of the first term (x) and the base of the last term(b/2a)

Add these two and raise at the second.

Then, square on the right side to get (b2) /(4a2)

quadratic-formula-proof-image

Get the same denominator on the right side:

quadratic-formula-proof-image

quadratic-formula-proof-image

Now, take the square root of each side:

quadratic-formula-proof-image

Simplify the left:

quadratic-formula-proof-image

quadratic-formula-proof-image

Rewrite the right side:

quadratic-formula-proof-image

Subtract (b) / 2 (a) of both sides:

quadratic-formula-proof-image

quadratic-formula-proof-image

Adding the numerator, keeping the same denominator, we get the quadratic formula:

quadratic-formula-proof-image

The + – between the b and the sign of the square root mean more or negative. In other words, most of the time, you will get two answers using the quadratic formula.

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