Question

How do you solve `3y^2-12=0`?

Answer 1

`y = +- 2`

Explanation:

It is necessary to manipulate the equation so that the unknown variable (here, `y`) is on its own on the left hand side of the equation. It will be equal to whatever is on the right hand side of the equation.

To make a start, by inspection, it will be necessary to divide `3 y^2` by `3` to remove the coefficient (and leave just `y^2`). It is necessary to do the same thing to both sides of the equation, so every term must be divided by `3`.

That is,

`3y^2 - 12 = 0`

implies

`(3y^2)/3 - 12/3 = 0/3`

that is

`y^2 - 4 = 0`

This may be rearranged (by adding `4` to both sides of the equation) to yield

`y^2 = 4`

To retrieve `y` on its own, it is necessary to take the square root of both sides. As `4` is a perfect square, this will be easy but take care! Remember that square numbers have two roots, a positive one and a negative one.

So,

`y^2 = 4`

implies

`sqrt(y^2) = sqrt(4)`

that is

`y = 2`
or
`y = -2`

Answer 2

`y = +-2`

Explanation:

`3y^2-12=0`

Multiply `3 xx (-12)` to get `-36` and use that to find two factors that when multiplied give `-36` and when added give `0`

`3y^2+6y-6y-12=0`

`3y(y+2)-6(y+2)=0`

Pull out the factors of the equation `3y^2-12=0`

`(3y-6) (y+2) = 0`

So

`3y-6=0 => y+2=0`

`3y=6 => y=-2`

Answer 3

`y = +-2`

Explanation:

This is a Difference of Two Squares problem -- with a slight disguise because it is multiplied by `3`

The Difference of Two Squares is a case of Special Factoring.

By memorization, the Difference of Two Squares factors like this:

`a^2 - b^2=(a + b)(a - b)`

`color(white)(mmmmmmmm)`――――――――

Given   `3y^2−12=0`    Solve for `y`

1) Factor out the 3 to see the perfect squares
`3 (y^2 - 4) = 0`

2) Factor the Difference of the Two Squares
`3 (y+2)(y-2)= 0`

3) Set the factors equal to zero and solve for `y`

`3= 0` `larr` discarded solution

`y + 2 = 0`
`y = -2` `larr` one answer

`y - 2 = 0`
`y = 2` `larr` the other answer

`color(white)(mmmmmmmm)`――――――――

Here's a video you can watch to see more about Special Factoring