Question

What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?

• A. -8
• B. 8
• C. 16
• D. -16

Answer 1

The Correct Option is A

To find the sum of all roots of the given equation, let's solve it step by step.
The given equation is: |x + 4|² - 10|x + 4| = 24
Let's substitute y = |x + 4|, so the equation becomes: y² - 10y = 24
Now, let's solve this quadratic equation for y: y² - 10y - 24 = 0
We can factor this quadratic equation: (y - 12)(y + 2) = 0
So, the possible values of y are y = 12 and y = -2.
However, since y represents the absolute value of (x + 4), it cannot be negative.
Therefore, we consider only y = 12.
Now, let's substitute y back in terms of |x + 4|: |x + 4| = 12
This leads to two cases: 1. x + 4 = 12 2. x + 4 = -12
Solving for x in both cases: 1. x = 12 - 4 = 8 2. x = -12 - 4 = -16
The sum of all roots (x values) is 8 + (-16) = -8.
Therefore, the sum of all roots of the given equation is -8.

Answer 2

|x + 4|² - 10|x + 4| = 24
Let u = |x + 4| and then replace |x + 4| with u, we get u² - 10u = 24
Subtract 24 from both sides to get: u² - 10u - 24 = 0
Factor to get: (u - 12)(u + 2) = 0
So, u = 12 or u = -2
Now Replace u with |x + 4| i.e. |x + 4| = 12 or |x + 4| = -2
If |x + 4| = 12, then x = 8 or -16
If |x + 4| = -2, then there are No Solutions, since |x + 4| will always be greater than or equal to zero.
So, there are only 2 solutions: x = 8 and x = -16
So, Sum of all possible solutions : x = 8 + (-16) = -8