Question

How do you solve `x^2 + 5x - 2 = 0`?

Answer 1

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

Explanation:

The difference of squares identity can be written:

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

We can solve the given quadratic by completing the square and using the difference of squares identity with `a=(2x+5)` and `b=sqrt(33)`.

I will multiply it by `4` to start with, to cut down on the need for arithmetic involving fractions:

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

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

`color(white)(0) = (2x)^2+2(2x)(5)+25-33`

`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))`

`color(white)(0) = 4(x+5/2-sqrt(33)/2)(x+5/2+sqrt(33)/2)`

Hence:

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