Question

How do you factor and solve `x^2-12x=-6`?

Answer 1

Bring over the `-6` and factor as you would a simple trinomial.

In this case, that doesn't work (it's "unfactorable"), and you have to use the quadratic formula to solve.

As a result, you'll get `x~=11.477` and `x~=0.523`.

Explanation:

Factoring

In order to factor, let's first bring the `-6` over, and equate the equation to `0`.

`x^2 - 12x = - 6`

`x^2 - 12x + 6= 0`

Now, let's factor it as you would a simple trinomial. Meaning, "what two numbers multiplied equals `ac` and added equals `b`?"

There are no numbers that fits this requirement (^). Therefore, it is "unfactorable". Due to this, we have to use the second way to factor: quadratic equation.

This ultimately brings us to...

Solving.

`x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}`

Now let's sub in the values.

`x=\frac{-(-12)\pm\sqrt{(-12)^2-4(1)(6)\ }}{2(1)}`

`x=\frac{12\pm\sqrt{120\ }}{2}`

At this point, we can solve for `x`. We will get two answers because of the `+-` sign.

`x=\frac{12\pm\sqrt{120\ }}{2}`

1. `x=\frac{12+\sqrt{120}}{2}`

`x~=\frac{22.954}{2}`

`x~=11.477`

2.
`x=\frac{12-\sqrt{120}}{2}`

`x~=\frac{1.046}{2}`

`x~=0.523`

We can graph the equation to check our work.

graph{x^2 - 12x + 6 [-2.8, 22.51, -6.33, 6.33]}

As you can see, the zeros match up.

Hope this helps :)