Question

If $2-i$ is a root of the equation $a{x}^2+12x+b=0$(where $a$ and $b$ are real), then the value of $ab$is

• A. $45$
• B. $15$
• C. $-15$
• D. $-45$
• E. $25$

Answer 1

The correct option is A

$45$

Explanation for the correct answer:

Step 1:Finding the other root:

If the roots of a quadratic polynomial are imaginary, then they exist in conjugate pairs, i.e., $a+ib$ and $a-ib$.

It is given that the root of the quadratic equation $a{x}^2+12x+b=0$ is $2-i$. Since the root is imaginary, the other root is the conjugate pair of $2-i$, i.e., $2+i$

Hence, the roots of the quadratic equation are $2+i$ and $2-i$.

Step 2 Inspecting the quadratic equation:

The sum of the roots of the quadratic equation can be given as $\frac{-12}{a}$.

$(2+i)+(2-i)=\frac{-12}{a}$

$a=-3$

The product of the roots of the quadratic equation can be given as $\frac{b}{a}$.

$(2+i)(2-i)=\frac{b}{a}$

$(4+1)=\frac{b}{-3}$

$b=-15$

Step 3: Finding the value of $ab$

The value of $ab$ can be calculated as shown below:

$ab=(-3)\times (-15)$

$ab=45$

Therefore the value of $ab$ is $45$.

Hence, Option (A) is correct.