Question

How do you solve `x^2+5x-6=0` by completing the square?

Answer 1

`x=1 or x=-6`

Explanation:

Here,
`x^2+5x-6=0`
...........................................................................................................................
To complete perfect square let `M` be the third term,
such that `x^2+5x +M` is a perfect square.

So,

`color(blue)((i)1^(st) term=x^2`
`color(blue)((ii)2^(nd)term=5x`
`color(blue)((iii)3^(rd)term=M`

Formula :
`color(blue)(3^(rd)term=(2^(nd)term)^2/(4xx 1^(st) term)`

`M=(5x)^2/(4xxx^2)=(25x^2)/(4x^2)=25/4`

`i.e. x^2+5x+(25/4)` is a perfect square.
......................................................................................................................
So,

`x^2+5x=6`

`=>x^2+5x+color(red)(25/4)=6+color(red)(25/4`

`=>(x+5/2)^2=(49)/4=(7/2)^2`

`=>x+5/2=+-7/2`

`=>x+5/2=7/2 or x+5/2=-7/2`

`=>x=7/2-5/2 or x=-7/2-5/2`

`=>x=2/2 or x=-12/2`

`=>x=1 or x=-6`