Determine Whether 3x² + 2√5x − 5 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 3x^2+2\sqrt5x-5=0 \]Solution
\[ a=3,\quad b=2\sqrt5,\quad c=-5 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(2\sqrt5)^2-4(3)(-5) \]
\[ D=20+60 \]
\[ D=80 \]
Since
\[ D>0 \]
the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{-2\sqrt5\pm\sqrt{80}}{6} \]
\[ x=\frac{-2\sqrt5\pm4\sqrt5}{6} \]
\[ x=\frac{2\sqrt5}{6}=\frac{\sqrt5}{3} \]
or
\[ x=\frac{-6\sqrt5}{6} =-\sqrt5 \]
Answer
\[
\boxed{x=\frac{\sqrt5}{3}\quad \text{or}\quad x=-\sqrt5}
\]
The equation has two distinct real roots.