Solve x² − (√2 + 1)x + √2 = 0 by Factorization
Question:
\[ x^2-(\sqrt2+1)x+\sqrt2=0 \]Solution
Given:
\[ x^2-(\sqrt2+1)x+\sqrt2=0 \]We need two numbers whose sum is \((\sqrt2+1)\) and product is \(\sqrt2\).
\[ \sqrt2+1=\sqrt2+1 \] \[ \sqrt2 \times 1=\sqrt2 \]Splitting the middle term:
\[ x^2-\sqrt2x-x+\sqrt2=0 \]Taking common factors:
\[ x(x-\sqrt2)-1(x-\sqrt2)=0 \] \[ (x-\sqrt2)(x-1)=0 \]Therefore,
\[ x-\sqrt2=0 \quad \text{or} \quad x-1=0 \] \[ x=\sqrt2 \quad \text{or} \quad x=1 \]