How do I complete the square?

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Answer 1 · 2015-10-23T20:16:56

#ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))#

The secret is that #b/(2a)# bit

Suppose you are given a quadratic equation to solve:

#2x^2-3x-2 = 0#

..which is in the form..

#ax^2+bx+c = 0# with #a = 2#, #b=-3# and #c=-2#

#b/(2a) = -3/4#

So we find:

#2(x-3/4)^2 = 2(x^2-(2*x*3/4)+(3/4)^2)#

#=2(x^2-(3x)/2+9/16)#

#=2x^2-3x+9/8#

So:

#2(x-3/4)^2-25/8 = 2(x-3/4)^2-9/8-2#

#=2x^2-3x+9/8-9/8-2#

#=2x^2-3x-2#

So:

#2x^2-3x-2 = 0#

turns into:

#2(x-3/4)^2-25/8 = 0#

Hence:

#(x-3/4)^2 = 25/16#

So:

#x-3/4 = +-sqrt(25/16) = +-5/4#

and

#x = 3/4+-5/4#