Solve the Following Quadratic Equation by Factorization
Question:
\[ 4\sqrt{3}x^2+5x-2\sqrt{3}=0 \]Solution
Given:
\[ 4\sqrt{3}x^2+5x-2\sqrt{3}=0 \]Product of the coefficient of \(x^2\) and the constant term:
\[ (4\sqrt{3})(-2\sqrt{3})=-24 \]We split the middle term \(5x\) as \(8x-3x\):
\[ 4\sqrt{3}x^2+8x-3x-2\sqrt{3}=0 \] \[ 4x(\sqrt{3}x+2)-\sqrt{3}(\sqrt{3}x+2)=0 \] \[ (\sqrt{3}x+2)(4x-\sqrt{3})=0 \]Therefore,
\[ \sqrt{3}x+2=0 \quad \text{or} \quad 4x-\sqrt{3}=0 \] \[ x=-\frac{2}{\sqrt{3}} =-\frac{2\sqrt{3}}{3} \] \[ x=\frac{\sqrt{3}}{4} \]