Quadratic Formula Proof

Solving and Deriving Problems with Quadratic Equations:

If the quadratic equations are simple, it's easy finding out the solution manually. But it's not the case with all quadratic equations. Most of them are so complicated that solving them without quadratic formula is very tough. To avoid quadratic formula problems, it was derived. Completing the squares is one of the method that can be used for derivation of quadratic formula. Proof of the quadratic formula using this method is given below. Consider the equation ax2 + bx + c =0, where a is a non-zero constant, b and c are constants. Dividing whole equation by 'a',

dividing quadratic equation

or

dividing quadratic formula

Now it's time we can apply completion of squares to get the proof of quadratic formula.

Rearranging and adding constants on both of its sides,

rearranging quadratic formula

i.e.

quadratic formula rearrangment

Taking square-root on both sides,

square root quadratic formula

or

quadratic formula square-root on both sides

This is the quadratic formula proof using completion of squares method. The formula can now be used for any quadratic equations. An alternate method was also pointed out by Larry Hoehn in 1975 for deriving the quadratic formula that can be accomplished in few steps. The proof of quadratic formula using Larry's method is given below considering same quadratic equation ax2 + bx + c =0. First step is to multiply both the sides by 4a, doing so, we have 4a2x2 + 4abx + 4ac =0.

After Rearranging, we have 4a2x2 + 4abx = -4ac, Now adding b2 on both sides,

4a2x2 + 4abx + b2= b2 - 4ac => (2ax+b)2= b2 - 4ac

Taking square-root on both the sides,

quadratic formula proof

After rearranging above equation, we have the final equation as

final equation of quadratic formula
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