The sum, of the squares of all the roots of the equation x²+|2 x-3|-4=0, is

Solution

Maths sir · 3 steps

Split at x = 3/2, solve each quadratic, and keep only the roots that sit in their own region.

1. Remove the modulus. |2x-3| changes form at x = 3/2. For x ≥ 3/2 the equation becomes x² + 2x - 7 = 0. For x < 3/2 it becomes x² - 2x - 1 = 0.

   Case	Root	Approx. value	Allowed?
   x ≥ 3/2	-1 + 2√2	1.83	yes
   x ≥ 3/2	-1 - 2√2	-3.83	no
   x < 3/2	1 + √2	2.41	no
   x < 3/2	1 - √2	-0.41	yes

2. Square the valid roots. (2√2 - 1)² = 8 - 4√2 + 1 = 9 - 4√2 and (1 - √2)² = 1 - 2√2 + 2 = 3 - 2√2.
3. So. The sum is 12 - 6√2 = 6(2 - √2), which is option (a).

The idea to keep. With a modulus, every case gives you candidate roots, not final ones. Always check each root against the condition of its own case before using it.

Answer from the NTA final answer key. Solution by PrepWiser.