Determine Whether √3x² + 10x − 8√3 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ \sqrt3x^2+10x-8\sqrt3=0 \]Solution
\[ a=\sqrt3,\quad b=10,\quad c=-8\sqrt3 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(10)^2-4(\sqrt3)(-8\sqrt3) \]
\[ D=100+96 \]
\[ D=196 \]
Since
\[ D>0 \]
the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{-10\pm14}{2\sqrt3} \]
\[ x=\frac{4}{2\sqrt3}=\frac{2}{\sqrt3} =\frac{2\sqrt3}{3} \]
or
\[ x=\frac{-24}{2\sqrt3} =-\frac{12}{\sqrt3} =-4\sqrt3 \]
Answer
\[
\boxed{x=\frac{2\sqrt3}{3}\quad \text{or}\quad x=-4\sqrt3}
\]
The equation has two distinct real roots.