Question

If $\frac{3}{2}+\frac{7}{2}i$ is a solution of the equation $a{x}^2-6x+b=0$, where $a$ and $b$ are real numbers, then the value of $a+b$ is

• A. $10$
• B. $22$
• C. $30$
• D. $29$
• E. $31$

Answer 1

The correct option is E

$31$

Explanation for the correct option:

Find the value of $a+b$:

Given, $\frac{3}{2}+\frac{7}{2}i$ is a solution of equation $a{x}^2-6x+b=0$

$\Rightarrow$ $a{\left(\frac{3}{2}+\frac{7}{2}i\right)}^2–\left(\frac{3}{2}+\frac{7}{2}i\right)+b=0$

$\Rightarrow$$a\left(-10+\frac{21}{2}\right)i–6\left(\frac{3}{2}+\frac{7}{2}i\right)+b=0$

$\Rightarrow$$10a–b=–9$ and $\left(\frac{21}{2}\right)a–21=0$

$\Rightarrow$$a=2$ and $b=29$

$\begin{array}{rcl}\therefore a+b& =& 2+29\\ & =& \mathbf{31}\end{array}$

Hence, Option ‘$E$’ is Correct.