Chapter 4.5, Problem 10PPSE

Solution

To solve: The equation $x^2+18=9x$ using factoring.

  $x=3$ , $x=6$ .

Given information:

The equation, $x^2+18=9x$ .

Concept used:

To factor an expression of the form $a{x}^2+bx+c$ , split the coefficient $b$ as sum of two real numbers such that their product equals $ac$ .

Zero product property: If $ab=0$ , then either $a=0$ or $b=0$ , or both $a=0$ , $b=0$ .

Calculation:

Rewrite the equation:

  $\begin{array}{c}{x}^2+18=9x\\ x^2+18-9x=9x-9x\\ x^2-9x+18=0\end{array}$

Factor the expression on the left of the equation and then apply zero product property find the solution.

  $\begin{array}{c}{x}^2-9x+18=0\\ \left(x^2-3x\right)+\left(-6x+18\right)=0\\ x\left(x-3\right)-6\left(x-3\right)=0\\ \left(x-3\right)\left(x-6\right)=0\\ \begin{array}{ccc}x-3=0& \text{or}& x-6=0\\ x=3& \text{or}& x=6\end{array}\end{array}$

Check:

To check the answers are actually a solution, substitute the values in the equation and observe the result. If the result is true, the solution is correct else incorrect.

Check $x=3$ :

  $\begin{array}{c}{\left(3\right)}^2+18=9\cdot 3\\ 9+18=27\\ 27=27\end{array}$

The result $27=27$ is true. So, $x=3$ is a solution of the equation.

Check $x=6$ :

  $\begin{array}{c}{\left(6\right)}^2+18=9\cdot 6\\ 36+18=54\\ 54=54\end{array}$

The result $54=54$ is true. So, $x=6$ is a solution of the equation.

Conclusion:

The solutions of the equation are $x=3$ and $x=6$ .