Question

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

Answer 1

`3(x+5/6)^2-49/12=y`

Explanation:

The goal of completing the square is to convert the equation into a perfect square trinomial. The benefit of this form is to be able to identify the vertex of a parabola very easily.

Factor out a 3

`3(x^2+5/3x)=2`

Take the coefficient from the x term and divide it by 2 and then square it.

`((5/3)/2)^2=(5/3*1/2)^2=(5/6)^2=25/36`

Include `25/36` on the left and include `3(25/36)` on the right because we factored out a 3 on the left in an earlier step.

`3(x^2+5/3x+25/36)=2+3(25/36)`

We now have a perfect square trinomial that can be written in a more compact form

`3(x+5/6)^2=2+3(25/36)`

Simplify

`3(x+5/6)^2=2+cancel3(25/(cancel36 12))`

`3(x+5/6)^2=2+(25/(12))`

Convert `2` to `24/12` so that we have common denominator.

`3(x+5/6)^2=24/12+(25/(12))`

`3(x+5/6)^2=49/12`

Subtract `49/12`

`3(x+5/6)^2-49/12=0`

`3(x+5/6)^2-49/12=y`

For more information please see the video tutorials below.