Solve the Following Quadratic Equation by Factorization
Question:
\[ \sqrt{2}x^2-3x-2\sqrt{2}=0 \]Solution
Given:
\[ \sqrt{2}x^2-3x-2\sqrt{2}=0 \]Product of the coefficient of \(x^2\) and the constant term:
\[ (\sqrt{2})(-2\sqrt{2})=-4 \]We split the middle term \(-3x\) as \(-4x+x\):
\[ \sqrt{2}x^2-4x+x-2\sqrt{2}=0 \] \[ \sqrt{2}x(x-2\sqrt{2})+\frac{1}{\sqrt{2}}(x-2\sqrt{2})=0 \] \[ (x-2\sqrt{2})\left(\sqrt{2}x+\frac{1}{\sqrt{2}}\right)=0 \]Multiplying the second factor by \(\sqrt{2}\):
\[ (x-2\sqrt{2})(2x+1)=0 \]Therefore,
\[ x-2\sqrt{2}=0 \quad \text{or} \quad 2x+1=0 \] \[ x=2\sqrt{2} \] \[ x=-\frac{1}{2} \]