How do you solve using the completing the square method x^2 = (3/4)x - (1/8)?

1 Answer
George C.
Mar 19, 2016

Multiply by 64 first to cut down on the fractions, then complete the square and use the difference of squares identity to find:

x = 1/2 or x=1/4

Explanation:

The difference of squares identity can be written:

A^2-B^2 = (A-B)(A+B)

I use this below, with A=(8x-3) and B=1.

color(white)()
To match ax^2+bx with a square, you normally look at:

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

In our example, we can rearrange the original equation to get one involving a=1 and b=-3/4, which would lead us to looking at:

(x-3/8)^2 = x^2-(3/4)x+9/64

These fractions get a little painful, so let us multiply the original equation by 64 before we start and cut down on the fractions...

x^2=(3/4)x-(1/8)

becomes:

64x^2=48x-8

which we can rearrange as:

0 = 64x^2-48x+8

=(8x)^2-2(8x)(3)+8

= (8x-3)^2-3^2+8

= (8x-3)^2-9+8

= (8x-3)^2-1

= (8x-3)^2-1^2

= ((8x-3)-1)((8x-3)+1)

= (8x-4)(8x-2)

= (4(2x-1))(2(4x-1))

= 8(2x-1)(4x-1)

So x=1/2 or x=1/4

Related questions
See all questions in Completing the Square
Impact of this question
1448 views around the world
You can reuse this answer
Creative Commons License