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

Question

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

Accepted Answer

Case $x \ge \tfrac32$: $|2x-3|=2x-3$.
$$x^2+2x-3-4=0 \implies x^2+2x-7=0 \implies x=-1\pm2\sqrt2$$
Only $x=2\sqrt2-1 \approx 1.83$ satisfies $x\ge\tfrac32$.

Case $x < \tfrac32$: $|2x-3|=3-2x$.
$$x^2+3-2x-4=0 \implies x^2-2x-1=0 \implies x=1\pm\sqrt2$$
Only $x=1-\sqrt2 \approx -0.41$ satisfies $x<\tfrac32$.

Sum of squares:
$$(2\sqrt2-1)^2 + (1-\sqrt2)^2 = (9-4\sqrt2) + (3-2\sqrt2) = 12-6\sqrt2 = 6(2-\sqrt2)$$

Solutions · 1