Question

The sum of squares of roots of $ |x - 2|^2 - |x - 2| - 2 = 0 $ and $ x^2 - 2|x - 3| - 5 = 0 $ equals to:

Hint:

For equations involving absolute values, break them into cases and consider both the positive and negative possibilities to find all roots.

Answer 1

Correct Answer: 43

We are given two equations:
1. \( |x - 2|^2 - |x - 2| - 2 = 0 \) 
2. \( x^2 - 2|x - 3| - 5 = 0 \) 
Step 1: Solve the first equation The first equation is: \[ |x - 2|^2 - |x - 2| - 2 = 0 \] Let \( y = |x - 2| \).
The equation becomes: \[ y^2 - y - 2 = 0 \] Factoring the quadratic equation: \[ (y - 2)(y + 1) = 0 \] Thus, \( y = 2 \) or \( y = -1 \).
Since \( y = |x - 2| \geq 0 \), we discard \( y = -1 \) and keep \( y = 2 \).
Therefore: \[ |x - 2| = 2 \] This gives two solutions: \[ x - 2 = 2 \quad \Rightarrow \quad x = 4 \] or \[ x - 2 = -2 \quad \Rightarrow \quad x = 0 \] So, the roots of the first equation are \( x = 4 \) and \( x = 0 \). 
Step 2: Solve the second equation The second equation is: \[ x^2 - 2|x - 3| - 5 = 0 \] Let \( z = |x - 3| \).
The equation becomes: \[ x^2 - 2z - 5 = 0 \quad \Rightarrow \quad z = \frac{x^2 - 5}{2} \] Substitute \( z = |x - 3| \) into the equation: \[ |x - 3| = \frac{x^2 - 5}{2} \] Now, solve this equation for roots by considering the two cases of the absolute value: Case 1: \( x - 3 = \frac{x^2 - 5}{2} \) \[ x - 3 = \frac{x^2 - 5}{2} \] Multiply both sides by 2: \[ 2(x - 3) = x^2 - 5 \] \[ 2x - 6 = x^2 - 5 \] Rearrange: \[ x^2 - 2x + 1 = 0 \] \[ (x - 1)^2 = 0 \] So, \( x = 1 \). Case 2: \( -(x - 3) = \frac{x^2 - 5}{2} \) \[ -(x - 3) = \frac{x^2 - 5}{2} \] Multiply both sides by 2: \[ -2(x - 3) = x^2 - 5 \] \[ -2x + 6 = x^2 - 5 \] Rearrange: \[ x^2 + 2x - 11 = 0 \] Use the quadratic formula: \[ x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-11)}}{2(1)} \] \[ x = \frac{-2 \pm \sqrt{4 + 44}}{2} \] \[ x = \frac{-2 \pm \sqrt{48}}{2} \] \[ x = \frac{-2 \pm 4\sqrt{3}}{2} \] \[ x = -1 \pm 2\sqrt{3} \] Thus, the roots of the second equation are \( x = 1 \), \( x = -1 + 2\sqrt{3} \), and \( x = -1 - 2\sqrt{3} \). 
Step 3: Calculate the sum of squares of roots The roots of the first equation are \( x = 4 \) and \( x = 0 \).
Their squares are: \[ 4^2 + 0^2 = 16 \] The roots of the second equation are \( x = 1 \), \( x = -1 + 2\sqrt{3} \), and \( x = -1 - 2\sqrt{3} \).
Their squares are: \[ 1^2 + (-1 + 2\sqrt{3})^2 + (-1 - 2\sqrt{3})^2 \] \[ = 1 + (1 - 2\sqrt{3} + 12) + (1 + 2\sqrt{3} + 12) \] \[ = 1 + 13 + 13 = 27 \] Thus, the sum of squares of all roots is: \[ 16 + 27 = 43 \]