+148 Given that the absolute value of the difference of the two roots of $ax^2 + 5x - 3 = 0$ is $\frac{\sqrt{61}}{3}$, and a is positive, what is the value of a?
+15000 Given that the absolute value of the difference of the two roots of \(ax^2+5x-3=0\) is \(\sqrt{\frac{61}{3}}\) , and a is positive, what is the value of a?
Hello powclaire!
\(\dfrac{\sqrt{61}}{3} =\dfrac{-5+ \sqrt{25+12a}}{2a} -\dfrac{-5- \sqrt{25+12a}}{2a}\\ =\dfrac{-5+ \sqrt{25+12a}+5+\sqrt{25+12a}}{2a} \)
\(\dfrac{\sqrt{25+12a}}{a}=\dfrac{\sqrt{61}}{3}\\ \dfrac{25+12a}{a^2}=\dfrac{61}{9}\)
\(61a^2-108a+225=0 \)
\(a=3\)
!
