Question

If one root of the equation $x^2+ax+12=0$ is $4$ while the equation $x^2+ax+b=0$ has equal roots, then the value of $b$ is

• A. $\frac{4}{49}$
• B. $\frac{49}{4}$
• C. $\frac{7}{4}$
• D. $\frac{4}{7}$

Answer 1

The correct option is B

$\frac{49}{4}$

Explanation for the correct option.

Step 1: Calculation for the value of $\mathit{a}$.

As the root of the equation $x^2+ax+12=0$ is $4$ then to find the value of the $a$ put $x=4$ in the same equation,

$\begin{array}{rcl}\therefore {\left(4\right)}^2+4a+12& =& 0\\ & \Rightarrow & 28+4a=0\\ & \Rightarrow & a=-7\end{array}$

Apply this value to the equation $x^2+ax+b=0$.

Step 2: Calculation for the value of $\mathit{b}$.

To find the value of $b$ we have the equation $x^2-7x+b=0$ with equal roots.

For the equal roots, we know that the discriminant $\left(D\right)$ is equal to $0$.

$\mathbf{\therefore }\mathit{D}\mathbf{=}{\mathit{B}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathit{A}\mathit{C}$, where $B=-7,A=1,\text{and}C=b$

$\begin{array}{rcl}\therefore B^2-4AC& =& 0\\ & \Rightarrow & {\left(-7\right)}^2-4\left(1\right)\left(b\right)=0\\ & \Rightarrow & 4b=49\\ & \Rightarrow & b=\frac{49}{4}\end{array}$

Hence, the correct option is (B).