Let R and let the equation E be |x|^2-2|x|+|-3|=0.

Question

Let $\lambda\in\mathbb R$ and let the equation $E$ be $|x|^2-2|x|+|\lambda-3|=0$. Then the largest element in the set $S=\{x+\lambda: x 	ext{ is an integer solution of } E\}$ is

Answer: 5

Accepted Answer

Rewrite: $(|x|-1)^2+|\lambda-3|=1$.

At $\lambda=3$: $(|x|-1)^2=1 \implies |x|=0,2 \implies x=0,\pm2$.

At $\lambda=4$ (or $\lambda=2$): $(|x|-1)^2=0 \implies |x|=1 \implies x=\pm1$.

The largest value of $x+\lambda$ among these cases is at $\lambda=4,\ x=1$: $x+\lambda=5$.

Solutions · 1