Question

$\frac{3x^2+1}{x^2-6x+8}$is equal to

• A. $3+\frac{49}{2\left(x-4\right)}-\frac{13}{2\left(x-2\right)}$
• B. $\frac{49}{2\left(x-4\right)}-\frac{13}{2\left(x-2\right)}$
• C. $-\frac{49}{2\left(x-4\right)}+\frac{13}{2\left(x-2\right)}$
• D. none of these

Answer 1

The correct option is A

$3+\frac{49}{2\left(x-4\right)}-\frac{13}{2\left(x-2\right)}$

Explanation for the correct option:

Step-1 Simplify the given data.

Given, $\frac{3x^2+1}{x^2-6x+8}$

After dividing we get quotient $3$ and remainder $18x-23$

$$\begin{gathered} \frac{3x^2+1}{x^2-6x+8}=3+\frac{18x-23}{x^2-4x-2x+8} \\ \mathrm{Let},\frac{18\mathrm{x}-23}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)}=\frac{\mathrm{A}}{\mathrm{x}-4}+\frac{\mathrm{B}}{\mathrm{x}-2}..........\left(i\right) \\ \Rightarrow \frac{18\mathrm{x}-23}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)}=\frac{\mathrm{A}\left(\mathrm{x}-2\right)+\mathrm{B}\left(\mathrm{x}-4\right)}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)} \\ \Rightarrow \frac{18\mathrm{x}-23}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)}=\frac{\left(\mathrm{A}+\mathrm{B}\right)\mathrm{x}-\left(2\mathrm{A}+4\mathrm{B}\right)}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)} \end{gathered}$$

Step-2: Compare the coefficients and solve the equations.

Compare, coefficients

$$\begin{gathered} A+B=18..........\left(ii\right) \\ 2A+4B=23...........\left(iii\right) \end{gathered}$$

From equation (i)

$A=18-B..........\left(iv\right)$

Put the value of $A$ in equation (ii)

$$\begin{gathered} 2\left(18-B\right)+4B=23 \\ \Rightarrow 36-2B+4B=23 \\ \Rightarrow B=-\frac{13}{2} \end{gathered}$$

Put value of $B$ in equation (iv)

$$\begin{gathered} A=18-\left(-\frac{13}{2}\right) \\ \Rightarrow A=\frac{49}{2} \end{gathered}$$

Step-3: Substitute the values.

Put the value of $A\&B$ in equation (i)

$$\begin{gathered} \Rightarrow \frac{18\mathrm{x}-23}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)}=\frac{\left({\displaystyle \frac{49}{2}}\right)}{\mathrm{x}-4}+\frac{\left(-{\displaystyle \frac{13}{2}}\right)}{\mathrm{x}-2} \\ \Rightarrow \frac{18\mathrm{x}-23}{\left(\mathrm{x}-4\right)\left(\mathrm{x}-2\right)}=\frac{49}{2\left(x-4\right)}-\frac{13}{2\left(x-2\right)} \end{gathered}$$

Now,

$\frac{3x^2+1}{x^2-6x+8}=3+\frac{49}{2\left(x-4\right)}-\frac{13}{2\left(x-2\right)}$

Therefore, correct answer is option $\left(A\right)$