Question

How do you solve `x^2+30x-7=0` by completing the square?

Answer 1

`x = 0.2315 or x = -30.2315`

Explanation:

The method using completing the square is based on:

`(x-y)^2 = x^2 - 2xy +y^2`

`(x-6)^2 = x^2 color(red)(-12)x +color(blue)(36)` Note that: `(color(red)(-12)/2)^2 = color(blue)(36)`
This relationship always exists in squaring a binomial.

`x^2 +30x-7 =0" 7 is not the correct constant"`

Move the 7 to the other side and add in the correct constant on both sides.
`x^2 color(red)(+30)x color(blue)(+225)= 7 color(blue)(+225)" "(color(red)(30)/2)^2 = color(blue)225`
`(x+15)^2 = 232" "x^2 +30x+225" is a square"`

`x + 15 = +-sqrt232 " square root both sides"`

Solve for x twice:

`x = +sqrt232 -15 = 0.2315`

`x = - sqrt232 -15 =-30.2315`