Determine Whether 2x² + 5√3x + 6 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 2x^2+5\sqrt3x+6=0 \]Solution
\[ a=2,\quad b=5\sqrt3,\quad c=6 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(5\sqrt3)^2-4(2)(6) \]
\[ D=75-48 \]
\[ D=27 \]
Since
\[ D>0 \]
the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{-5\sqrt3\pm\sqrt{27}}{4} \]
\[ x=\frac{-5\sqrt3\pm3\sqrt3}{4} \]
\[ x=\frac{-2\sqrt3}{4} =-\frac{\sqrt3}{2} \]
or
\[ x=\frac{-8\sqrt3}{4} =-2\sqrt3 \]
Answer
\[
\boxed{x=-\frac{\sqrt3}{2}\quad \text{or}\quad x=-2\sqrt3}
\]
The equation has two distinct real roots.