Question

How do you solve `|x ^ { 2} - 10x | = 24`?

Answer 1

Please see below.

Explanation:

There are two numbers whose absolute value (modulus) is `24`.

They are `24` and `-24`.

So we will solve two equations. We'll solve

`x^2-10x = 24`

and we shall solve `x^2-10x = -24`

`x^2-10x = 24` has exponent `2` on the unknown. That makes it a quadratic equation. To solve it we want `0` on the left.

`x^2-10x-24 = 0`

I prefer to solve by factoring when I can, but you might prefer to use the formula.

We need two numbers that multiply to give us `-24` and add to give us `-10`.
Because we want a negative product, one of the numbers is positive and the other is negative.
Because the sum is negative (`-10`) the larger number is negative.
Look at the possibilities:

`1 xx -24` has a sum of `-23` -- not what we want
`2 xx -12` has a sum of `-10` -- that's it!

We can factor `2^2-10x-24 = (x+2)(x-12)` (check by multiplying)

`(x+2)(x-12) = 0`

`x+2 = 0` or `x-12 = 0`

`x = -2` or `x = 12`

Now solve `x^2-10x=-24`

`x^2-10x+24 = 0`

We need two numbers that multiply to give us `24` and add to give us `-10`.
Because we want a positive product, either both numbers are positive or both are negative.
Because the sum is negative (`-10`) the numbers we want are both negative.
Look at the possibilities:

`-1 xx -24` has sum `-25`
`-2 xx -12` has sum `-14`
`-3 xx -8` has sum `-11`
`-4 xx -6` has sum `-10` -- that's the one we want

We can factor `2^2-10x+24 = (x-4)(x-6)` (check by multiplying)

`(x-4)(x-6) = 0`

`x-4 = 0` or `x-6 = 0`

`x = 4` or `x = 6`

The solutions are: `-2`, `12`, `4`, and `6`