Question

How do you solve `-3x^2+7x= -5` by completing the square?

Answer 1

`x=(7+-sqrt109)/6`

Explanation:

You should always start by what you want to reach, namely an expression on the form of
`(x-a)^2=b`
But first, to get some idea, let's start with a graph of the function
`f(x)= -3x^2+7x+5`:

We want `(x-a)^2=x^2-2ax+a^2` on the left side
Our expression has
`-3x^x+7x=-5`
Let's simplify by dividing the whole on `-3`:
`x^2-7/3x=5/3`

Therefore `-2ax=-7/3x` or `2a=7/3`, `a=7/6`
Add `a^2=(7/6)^2` on both sides:
`x^2-7/3x+(7/6)^2=5/3+49/6^2`
`(x-7/6)^2=(5*12+49)/6^2=109/6^2`
`x-7/6=+-sqrt109/6`
`x=(7+-sqrt109)/6`