Question

How do you factor `(m-7)= -(m-7)^2 + 3`?

Answer 1

`m = 13/2+-sqrt(13)/2`

or

`(m - 13/2 - sqrt(13)/2)(m - 13/2 + sqrt(13)/2) = 0`

Explanation:

Given:

`(m-7) = -(m-7)^2+3`

First add `(m-7)^2-3` to both sides to get:

`(m-7)^2+(m-7)-3 = 0`

This is in the form:

`ax^2+bx+c = 0`

with `x = (m-7)`, `a=color(purple)(1)`, `b=color(blue)(1)` and `c = color(brown)(-3)`

So we can use the quadratic formula to find:

`m-7 = (-b+-sqrt(b^2-4ac))/(2a)`

`color(white)(m-7) = (-color(blue)(1)+-sqrt(color(blue)(1)^2-4(color(purple)(1))(color(brown)(-3))))/(2(color(purple)(1)))`

`color(white)(m-7) = (-1+-sqrt(1+12))/2`

`color(white)(m-7) = -1/2+-sqrt(13)/2`

Add `7` to both ends to find:

`m = 13/2+-sqrt(13)/2`