Solve the Following Quadratic Equation by Factorization
Question:
\[ x^2-(\sqrt{3}+1)x+\sqrt{3}=0 \]Solution
Given:
\[ x^2-(\sqrt{3}+1)x+\sqrt{3}=0 \]We need two numbers whose product is \(\sqrt{3}\) and sum is \(\sqrt{3}+1\).
\[ \sqrt{3}\times 1=\sqrt{3} \] \[ \sqrt{3}+1=\sqrt{3}+1 \]Splitting the middle term:
\[ x^2-\sqrt{3}x-x+\sqrt{3}=0 \]Taking common factors:
\[ x(x-\sqrt{3})-1(x-\sqrt{3})=0 \] \[ (x-\sqrt{3})(x-1)=0 \]Therefore,
\[ x-\sqrt{3}=0 \quad \text{or} \quad x-1=0 \] \[ x=\sqrt{3} \quad \text{or} \quad x=1 \]