Chapter 12.5, Problem 25WE

To determine

Solve the quadratic equation.

Answer to Problem 25WE

  $x\in \left\{4,\frac{32}{9}\right\}$

Explanation of Solution

Given:

The equation is :

  $9{(x-4)}^2+4(x-4)=0$

Calculation:

The given equation $9{(x-4)}^2+4(x-4)=0$ .

Substitute x - 4 = y

So,

  $\begin{array}{l}$ 9{(x-4)}^2+4(x-4)=0$\Rightarrow 9y^2+4y=0\\ \Rightarrow y(9y+4)=0\\ \Rightarrow y=0\text{ or 9}y+4=\mathrm{0................}[\text{zero product property]}\\ \Rightarrow y=0\text{ or }y=-\frac{4}{9}\\ \Rightarrow x-4=0\text{ or x-4}=-\frac{4}{9}\mathrm{...............}[y=x-4]\\ \Rightarrow x=4\text{ or x}=4-\frac{4}{9}\\ \Rightarrow \Rightarrow x=4\text{ or x}=\frac{32}{9}\end{array}$

Hence , $x\in \left\{4,\frac{32}{9}\right\}$