Q. 3 - Precalculus Enhanced with Graphing Utilities

Question

find the real solutions of each equation.

$x^2-8x-9=0$

Step-by-Step Solution

Answer

The solutions of the equation $x^2-8x-9=0$ are

$x=-1 \& 9$

Step 1. Given data

The given equation is 

$x^2-8x-9=0$

Step 2. use Factorisation

Factorize the equation 

$$\begin{gathered} x^2-8x-9=0 \\ {x}^2-9x+x-9=0 \\ x(x-9)+1(x-9)=0 \\ (x+1)(x-9)=0 \end{gathered}$$

Step 3. Apply product property

Use zero product property and consider

 $$\begin{gathered} (x+1)=0 \\ x=-1 \end{gathered}$$

and$$\begin{gathered} x-9=0 \\ x=9 \end{gathered}$$

solutions of the equation are $x=-1 \& 9$

Step 4. Verification

Substitute $-1$ for x in the equation 

$$\begin{gathered} x^2-8x-9=0 \\ {(-1)}^2-8(-1)-9\stackrel{?}{=}0 \\ 1+8-9\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$

Substitute $9$ for x in the equation 

$$\begin{gathered} x^2-8x-9=0 \\ {\left(9\right)}^2-8\left(9\right)-9\stackrel{?}{=}0 \\ 81-72-9\stackrel{?}{=}0 \\ 0=0 \end{gathered}$$

Both sides of the equation are the same for both solutions so the solution is correct.