Chapter 6.6, Problem 59S

To determine

To calculate: the value of $c(a-3)+d(a+1)$ .

Answer to Problem 59S

The value is $4\left(a^2-2a+4\right)$ .

Explanation of Solution

Given information:

  $\begin{array}{l}c(x)=x^2-2x\\ d(x)=3x^2-6x+4\end{array}$ .

Formula used:

Step 1. Write the original function.

Step 2. Replace the value of x respectively.

Step 3. Solve the equations.

Calculation:

Consider ,

  $\begin{array}{l}c(x)=x^2-2x\text{ (i)}\\ d(x)=3x^2-6x+4\text{ (ii)}\end{array}$

To evaluate $c(a-3)\text{ and }d(a+1)$ , replace the x in equation (i) and equation (ii) respectively.

  $c(a-3)={\left(a-3\right)}^2-2\left(a-3\right)$

  $\begin{array}{l}=a^2-6a+9-2a+6\\ =a^2-8a+15\text{ (iii)}\end{array}$

  $d(a+1)=3{\left(a+1\right)}^2-6\left(a+1\right)+4$

  $\begin{array}{l}=3(a^2+2a+1)-6a-6+4\\ =3a^2+6a+3-6a-2\\ =3a^2+1\text{ (iv)}\end{array}$

Now solving equation (iii) and (iv) we get,

  $\begin{array}{l}=a^2-8a+15+(3a^2+1)\\ =a^2-8a+15+3a^2+1\\ =4a^2-8a+16\\ =4\left(a^2-2a+4\right)\end{array}$

Hence, the value is $4\left(a^2-2a+4\right)$ .