What is the sum of all possible solutions of the equation

|x + 4|^{2} – 10|x + 4| = 24?

##### Title

Absolute quadratic

##### Your Result

Correct

##### Difficulty

Very Hard

##### Your Pace

0:01

##### Others' Pace

2:40

## Video Explanation

## Text Explanation

Here's the equation we're working with:

|x+4|` – 10|x+4| = 24`

We can make this simpler and easier to work with by temporarily replacing |x+4| with u. So:

|x+4| = u

u` – 10u = 24`

Now we can solve this equation for u. First we'll subtract 24 from both sides, and then we'll factor the left hand side.

u` – 10u = 24`

u` – 10u - 24 = 0`

(u – 12) (u + 2) = 0

From here, we can see that there are two answers:

u = –2

u = 12

Now, in each of these answers, we can change u back to |x+4|:

|x+4| = –2

|x+4| = 12

So we have two equations. Let's start by looking at the first one:

|x+4| = –2

This equation has no solution, because the absolute of any number can must be positive. So it is impossible for |x+4| to be equal to a negative number - and thus, we can conclude that there is no solution to this equation.

Next let's look at the second equation:

|x+4| = 12

From here, we can make two conclusions: either x + 4 = 12, or x + 4 = –12. Let's solve each of these for x.

x + 4 = 12

x = 8

And:

x + 4 = –12

x = –16

So we have two solutions - that is, two possible values for x: 8 and –16. The question asks us for the sum of the solutions, so:

8 + (–16) = –8

So the answer is D.

**FAQ: Why do we substitute in the U? Can't we solve without that?**

**A: **Yes, we can answer this question without using U substitution (also known as "the Something Method" in the related lesson video below). Without substituting, you can do these equations in the exact same way, but it's much more complicated looking and takes longer to write. We use the U just to make things easier to work with. Here is the solution without the U:

|x+4|^{2} – 10|x+4| = 24

|x+4|^{2} – 10|x+4| – 24 = 0

(|x+4| – 12) (|x+4| + 2) = 0

|x+4| = –2 (not possible!)

or

|x+4| = 12 (good)

x+4 = 12

x = 8

or

x+4 = –12

x = –16

It's the same process, right? It's just uglier! So why not use the U?

**FAQ: I don't understand why |x+4| = –2 is not possible? Why is there no solution for this expression?**

**A:** |x+4| = –2 is not possible because the absolute value of any number is always going to be **non-negative**. Here, the absolute value of the expression is equal to a *negative* number, which is *impossible* due to the absolute value bars.

## Related Lessons

Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.