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) $\varnothing$ Find all values of x satisfying the given conditions.

Question

Quizplus · Accepted Answer

The equation $\left| x ^ { 2 } - 4 x + 4 \right| = 2$ can be solved by considering both the positive and negative scenarios of the absolute value expression. The expression inside the absolute value, $x^2 - 4x + 4$, can be factored as $(x-2)^2$, which simplifies the equation to $\left| (x-2)^2 \right| = 2$. This leads to two cases: $(x-2)^2 = 2$ and $(x-2)^2 = -2$. The latter has no real solutions, so we focus on the former, which simplifies to $x-2 = \pm \sqrt{2}$, yielding solutions $x = 2 \pm \sqrt{2}$.

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

Solve the Absolute Value Equation or Indicate That the Equation $\left| x ^ { 2 } - 4 x + 4 \right| = 2$