Solve Equations Involving Absolute Value
Solve the absolute value equation or indicate that the equation has no solution.
-$$\left| x ^ { 2 } - 4 x + 4 \right| = 2$$
A) $\{ 2 - \sqrt { 2 } , 2 + \sqrt { 2 } \}$
B) $\{ 2 - \sqrt { 2 } \}$
C) $\{ 2 + \sqrt { 2 } \}$
D) $\varnothing$

Question

Quizplus · Accepted Answer

We can start by using the definition of absolute value: 
$$|x^2-4x+4|=\begin{cases}x^2-4x+4 & \text{if } x^2-4x+4\geq 0\ -(x^2-4x+4) & \text{if } x^2-4x+4<0\end{cases}$$
So we have two cases to consider:

Case 1: $x^2-4x+4\geq 0$

In this case, $|x^2-4x+4|=x^2-4x+4$, so we need to solve the equation $x^2-4x+4=2$. This gives us $x=2\pm\sqrt{2}$.

Case 2: $x^2-4x+4<0$

In this case, $|x^2-4x+4|=-(x^2-4x+4)$, so we need to solve the equation $-(x^2-4x+4)=2$. This gives us $x=2\pm\sqrt{2}$ (notice that we still get the same solutions).

Putting these solutions together, we obtain $\{2-\sqrt{2},2+\sqrt{2}\}$ as the solution set, which corresponds to choice A.

Solve Equations Involving Absolute Value Solve the Absolute Value Equation ∣x2−4x+4∣=2\left| x ^ { 2 } - 4 x + 4 \right| = 2​x2−4x+4​=2

Solve Equations Involving Absolute Value
Solve the Absolute Value Equation $\left| x ^ { 2 } - 4 x + 4 \right| = 2$