Determine Whether 3x² − 5x + 2 = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 3x^2-5x+2=0 \]Solution
\[ a=3,\quad b=-5,\quad c=2 \]
Find the discriminant:
\[ D=b^2-4ac \]
\[ D=(-5)^2-4(3)(2) \]
\[ D=25-24 \]
\[ D=1 \]
Since
\[ D>0 \]
the equation has two distinct real roots.
\[ x=\frac{-b\pm\sqrt{D}}{2a} \]
\[ x=\frac{5\pm1}{6} \]
\[ x=\frac{6}{6}=1 \]
or
\[ x=\frac{4}{6}=\frac{2}{3} \]
Answer
\[
\boxed{x=1\quad \text{or}\quad x=\frac{2}{3}}
\]
The equation has two distinct real roots.