Question

How do you solve `x^2=24x+10` by completing the square?

Answer 1

`color(green)(x=12-sqrt(154))color(white)("XX")` or `color(white)("XX")color(green)(x=12+sqrt(154)`

Explanation:

Given
`color(white)("XXX")x^2=24x+10`

Shift all the terms which include the variable `x` to the left side:
`color(white)("XXX")x^2-24x=10`

Since, in general, the expansion of a squared binomial has the structure:
`color(white)("XXX")(x+a)^2=underline(x^2+2ax) + a^2`

If `x^2-24x` are the first two terms of the expansion of a squared binomial
then the third terms should be `(-24/2)^2=(-12)^2=12^2`

In order to complete the square we will need to add `12^2` (to both sides)

`color(white)("XXX")x^2-24x+12^2=10+12^2`

Writing as a squared binomial and simplifying:
`color(white)("XXX")(x-12)^2=154`

Taking the square root of both sides
`color(white)("XXX")x-12=+-sqrt(154)`
Adding `12` to both sides
`color(white)("XXX")x=12+-sqrt(154)`

Answer 2

`x=12+-sqrt(154)`

Explanation:

`x^2-24x = 10`
take the factor of x, half it and square, then add both sides
`x^2-24x+144 = 10+144`
`(x-12)^2 = 154`
`sqrt((x-12)^2)=+-sqrt(154)`
`(x-12)=+-sqrt(154)`
`x=12+-sqrt(154)`