Question

If $x=\frac{2}{3}$ and $x=-3$ are the roots of the quadratic equation $a{x}^2+7x+b=0$ then find the values of a and b.

Answer 1

Given: $a{x}^2+7x+b=0$
Since, $x=\frac{2}{3}$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
$$\begin{gathered} a{\left(\frac{2}{3}\right)}^2+7\left(\frac{2}{3}\right)+b=0 \\ \Rightarrow \frac{4}{9}a+\frac{14}{3}+b=0 \\ \Rightarrow 4a+42+9b=0 \\ \Rightarrow 4a+9b=-42...\left(1\right) \end{gathered}$$

Since, $x=-3$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get
$$\begin{gathered} a{\left(-3\right)}^2+7\left(-3\right)+b=0 \\ \Rightarrow 9a-21+b=0 \\ \Rightarrow 9a+b=21...\left(2\right) \end{gathered}$$
From (1) and (2), we get
$a=3,b=-6$