Which of the following values of $$ x $$ will satisfy the equation $$ |x-1|^{2}+3|x-1|-4=0 $$ ?

Question

UpStudy AI · Accepted Answer

Let $$ y = |x-1| $$, so the equation becomes

$$
y^2 + 3y - 4 = 0.
$$

We solve this quadratic equation by using the quadratic formula:

$$
y = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm \sqrt{25}}{2}.
$$

This simplifies to

$$
y = \frac{-3 \pm 5}{2}.
$$

Thus, we have two potential solutions:

$$
y = \frac{-3 + 5}{2} = 1 \quad \text{or} \quad y = \frac{-3 - 5}{2} = -4.
$$

Since $$ y = |x-1| \geq 0 $$, we discard $$ y = -4 $$.

Now, we solve

$$
|x-1| = 1.
$$

This absolute value equation gives two cases:

1) $$ x-1 = 1 $$ leading to

$$
x = 2,
$$

2) $$ x-1 = -1 $$ leading to

$$
x = 0.
$$

Thus, the values of $$ x $$ that satisfy the equation are

$$
x = 0 \quad \text{and} \quad x = 2.
$$
The values of $$ x $$ that satisfy the equation are $$ x = 0 $$ and $$ x = 2 $$.

Which of the following values of \( x \) will satisfy the equation \( |x-1|^{2}+3|x-1|-4=0 \) ?