If x is a non-zero integer and |x^2 - 2| - |x - 2| = -2, what is the

Ahh that makes sense. Bunuel

But then that cross-checking for up to 5 values of x will increase solving time right?
Is there a faster way of doing it,
Coz I do the
solving for each combination of (++,--,+-,-+) and then re-substitution to check for validity seems so long­

\(|x^2 - 2| - |x - 2| = -2\)

Let's obtain the critical points of the equation.

• \(x^2 - 2 = 0\)
  
  • ⇒ \(x = \pm \sqrt{2}\)
• \(x - 2 = 0\)
  
  • ⇒ \(x = 2\)

­Let's plot these over a number line 

--- Region 1 --- \(-\sqrt{2}\)­ --- Region 2 --- \(\sqrt{2}\) --- Region 3 --- 2 --- Region 4 ---­

Region 1 (\(x < -\sqrt{2}\))

\(|x^2 - 2| - |x - 2| = -2\)

\(x^2 - 2 + x - 2 = -2\)

\(x^2 - 2 + x = 0\)

Value of x = -2 (we ignore x = 1, as the value falls out of range.)

Region 2 (\(-\sqrt{2} \leq x \leq \sqrt{2}\))­

\(|x^2 - 2| - |x - 2| = -2\)

\(-x^2 + 2 + x - 2 = -2 \)

\(-x^2 + x + -2 = 0\)

Between 1 and -1, -1 satisfies the equation. 

Region 3 (\(\sqrt{2} < x < 2\))

We can ignore this region, as there are no integers in this region.

Region 4 (\(x \geq 2\)­)­

\(|x^2 - 2| - |x - 2| = -2\)

\(x^2 - 2 - x + 2 = -2\)

\(x^2 - x + 2 = 0\)

No integer value of \(x\) satisfies the equation. Hence, we can ignore this region.

Hence, the only possible value of x = {-1, -2}

For both values of x, \(x^4 - 5x^2 + 4 = 0\)

Option C­

­Looking at \(|x^2−2|−|x−2|= −2\), to have a negative value from subtracting these two absolute values one needs \(|x−2|\) to be greater than \(|x^2−2|\). 

Looking at the two absolutes, one notices that:

1. When \(x = 1\) the two absolute values will be equal which in the equation above will result in an answer of \(0\). 

2. When \(x > 1\) then \(|x^2−2|\) >\(|x−2|\) and will yeild a positive number when subtracting.

3. When \(x ≤ -3\) then once again \(|x^2−2|\) >\(|x−2|\).

This leaves only \(-1\) and \(-2\), both of which are values for \(x\) with which  \(|x^2−2|−|x−2|= −2\).

Plugging these values into \(x^4 - 5x^2 + 4\):

[-1]: \(1 - 5 + 4 = 0\)

[-2]: \(16 - 20 + 4 = 0\)

ANSWER C


 ­

Bunuel, Is there any other way of solving this question ?

Algebraically, this is time consuming. Finding the roots of the equation, checking the values and then plugging in the value of x in the final equation is not the ideal method to approach the question, in my opinion.

BTW, the answer is C.­