If one root of the quadratic equation $2 x^{2} - 3 x + p = 0$ is 3. Find the other root of the quadratic equation. Also, find the value of $p$ .

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Solution

Step 1:
Find the value of $p$ :
Since, $3$ is a root of given quadratic equation. Then, $x = 3$ satisfies the given equation.
$2 x^{2} - 3 x + p = 0 \begin{array}{rcl} \Rightarrow & 2 {(3)}^{2} - 3 (3) + p = 0 \\ \Rightarrow & 18 - 9 + p = 0 \\ \Rightarrow & 9 + p = 0 \\ \Rightarrow & p = - 9 \end{array}$
Step 2:
Find the other root of given equation:
Substitute $p = - 9$ in given equation.
$2 x^{2} - 3 x - 9 = 0$
Use the factorization method to solve the obtained quadratic equation.
$2 x^{2} - 3 x - 9 = 0 \begin{array}{rcl} \Rightarrow & 2 x^{2} - 6 x + 3 x - 9 = 0 \\ \Rightarrow & 2 x (x - 3) + 3 (x - 3) = 0 \\ \Rightarrow & (2 x + 3) (x - 3) = 0 \\ \Rightarrow & 2 x + 3 = 0 or x - 3 = 0 \\ \Rightarrow & x = - \frac{3}{2} or x = 3 \end{array}$
Therefore, the other root of given equation is $x = - \frac{3}{2}$ and the value of $p = - 9$ .

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