Chapter 1.3, Problem 25AYU

In Problems 17-36, solve each equation algebraically. Verify your solution using a graphing utility.

$(x+7)(x-1)={(x+l)}^2$

To determine

To find: The solution for the equation $\text{(}x\text{ + 7) (}x\text{ - 1) = }{(x+1)}^2$ algebraically and verify using a graphing utility.

Answer to Problem 25AYU

Solution:

$x\text{ = 2}$

Explanation of Solution

Given:

The equation $\text{(}x\text{ + 7) (}x\text{ − 1) = }{(x+1)}^2$

Algebraic solution:

$\text{(}x\text{ + 7) (}x\text{ − 1) = }{(x+1)}^2$

$x^2\text{ − x + 7}x\text{ − 7 = }{x}^2\text{ + 2}x\text{ + 1}$

$x^2\text{ + 6}x\text{ − 7 = }{x}^2\text{ + 2}x\text{ + 1}$

$x^2\text{ + 6}x\text{ − 7 + }{x}^2\text{ = }{x}^2\text{ + 2}x\text{ + 1 − }{x}^2$Subtract $x^2$ from both sides

$\text{6}x\text{ − 7 = 2}x\text{ + 1}$

$\text{6}x\text{ - 7 − 2}x\text{ = 2}x\text{ + 1 − 2}x$Subtract $\text{2}x$ from both sides

$\text{4}x\text{ − 7 = 1}$

$\text{4}x\text{ − 7 + 7 = 1 + 7}$Add 7 to both sides

$\text{4}x\text{ = 8}$

$x\text{ = }\frac{\text{8}}{\text{4}}$Divide both sides by 4

$x\text{ = 2}$

Check:

Let $x\text{ = 2}$ in the expression in $y$ on the left side of the equation

$\left(x\text{ + 7}\right)\left(x\text{ − 1}\right)\text{ = }\left(\text{2 + 7}\right)\left(\text{2 − 1}\right)$

$\text{= 9 × 1}$

$\text{= 9}$

Let $x\text{ = 2}$ in the expression in $y$ on the right side of the equation

${(x+1)}^2={(2+1)}^2$

$=3^2$

$\text{= 9}$

Since the two expressions are equal, the solution $x\text{ = 2}$ checks.