Question

How do you solve using completing the square method `x^2+5x-2=0`?

Answer 1

`x = -5/2+-sqrt(33)/2`

Explanation:

The difference of square identity can be written:

`a^2-b^2 = (a-b)(a+b)`

Use this with `a=(2x+5)` and `b=sqrt(33)`

First premultiply by `4` to reduce the amount of fraction arithmetic we need to do...

`0 = 4(x^2+5x-2)`

`color(white)(0) = 4x^2+20x-8`

`color(white)(0) = (2x)^2+2(2x)(5)+5^2-33`

`color(white)(0) = (2x+5)^2-(sqrt(33))^2`

`color(white)(0) = ((2x+5)-sqrt(33))((2x+5)+sqrt(33))`

`color(white)(0) = (2x+5-sqrt(33))(2x+5+sqrt(33))`

So:

`2x = -5+-sqrt(33)`

So:

`x = -5/2+-sqrt(33)/2`