(x-4) |x-4| i am looking for the simplification of this. i think the answer should be x^2 - 16, my teacher gave us an answer, but i don't believe it is correct. she gave us…

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asked Jan 6, 2023 102k views

1 Answer

Malreddysid · Answer 1 · 2023-01-12T06:31:19+0000

It depends on the sign of
x-4 .

Recall the absolute value function's definition:

$|x| = \begin{cases}x & \text{if } x \ge 0 \\ -x & \text{if } x < 0\end{cases}$

Now, if
$x-4\ge0$ , then

(x-4)|x-4| = (x-4)(x-4) = x^2 - 8x + 16

On the other hand, if
x-4<0 , then

(x-4)|x-4| = -(x-4)(x-4) = -(x-4)^2 = -x^2 + 8x - 16

What you first suggested is incorrect. We have

x^2 - 16 = (x - 4) (x + 4)

but
$|x-4| \\eq x + 4$ .

We can show this using the definition again. If
$x-4\ge0$ , then

$x-4 = x+4 \implies -4=4$

which is a contradiction; on the other hand, if
x-4<0 , then

$-x+4 = x+4 \implies 2x = 0 \implies x =0$

holds for only one value of
.