Question:
If $x=\frac{2}{3}$ and $x=-3$ are the roots of the quadratic equation $a x^{2}+7 x+b=0$ then find the values of $a$ and $b$.
Solution:
Given: $a x^{2}+7 x+b=0$
Since, $x=\frac{2}{3}$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
$a\left(\frac{2}{3}\right)^{2}+7\left(\frac{2}{3}\right)+b=0$
$\Rightarrow \frac{4}{9} a+\frac{14}{3}+b=0$
$\Rightarrow 4 a+42+9 b=0$
$\Rightarrow 4 a+9 b=-42$ .....(1)
Since, $x=-3$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
$a(-3)^{2}+7(-3)+b=0$
$\Rightarrow 9 a-21+b=0$
$\Rightarrow 9 a+b=21$ ......(2)
From (1) and (2), we get
$a=3, b=-6$