Question

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

Answer 1

given  equation: $x^2+ \left | 2x-3 \right |-4=0$

To find : sum of the roots

using the quadratic formula first we find the roots of the equation

i.e

first case:

$x\geq \frac{3}{2}\\\\x^2+2x-3-4=0\\\\x^2+2x-7=0\\\\x= \frac{(-2\pm \sqrt{32})}{2}\\\\-1 \pm 2\sqrt{2}\\\\but,only \\\\-1+2\sqrt{2}>\frac{3}{2}$

second case:

$x<\frac{3}{2}\\\\x^2+3-2x-4=0\\\\x^2-2x-1=0\\\\x= \frac{(2\pm \sqrt{32})}{2}= 1\pm \sqrt{2}\\\\only \\\\1-\sqrt{2} <\frac{3}{2}$

thus , Sum of the roots is

$-1+2\sqrt{2} + (1 -\sqrt{2})= \sqrt{2}$

hence , the sum of the roots is √2