Determine Whether 3a²x² + 8abx + 4b² = 0 Has Real Roots and Find the Roots
Question
Determine whether the given quadratic equation has real roots and if so, find the roots:
\[ 3a^2x^2+8abx+4b^2=0,\quad a\ne0 \]Solution
\[ A=3a^2,\quad B=8ab,\quad C=4b^2 \]
Find the discriminant:
\[ D=B^2-4AC \]
\[ D=(8ab)^2-4(3a^2)(4b^2) \]
\[ D=64a^2b^2-48a^2b^2 \]
\[ D=16a^2b^2 \]
Since
\[ D=16a^2b^2 \ge 0 \]
the equation has real roots.
\[ x=\frac{-B\pm\sqrt{D}}{2A} \]
\[ x=\frac{-8ab\pm4ab}{6a^2} \]
\[ x=\frac{-4ab}{6a^2} =-\frac{2b}{3a} \]
or
\[ x=\frac{-12ab}{6a^2} =-\frac{2b}{a} \]
Answer
\[
\boxed{x=-\frac{2b}{3a}\quad \text{or}\quad x=-\frac{2b}{a}}
\]
The equation has two real roots.