Answer:
Area of the rectangle(A) is given by:
[tex]A=lw[/tex]
where,
l is the length and w is the width of the rectangle.
As per the statement:
The length and width of a rectangular patio are, (x + 8) feet and (x + 6) feet, respectively. If the area of the patio is 160 square feet
⇒A = 160 square feet, l= x+8 feet and w = x+6 feet
then;
[tex](x+8)(x+6) = 160[/tex]
⇒[tex]x^2+6x+8x+48 = 160[/tex]
⇒[tex]x^2+14x+48 = 160[/tex]
Subtract 48 from both sides we have;
[tex]x^2+14x=112[/tex]
Using completing square method:
[tex]x^2+14x+7^2=112+7^2[/tex]
⇒[tex](x+7)^2=112+49[/tex]
⇒[tex](x+7)^2=161[/tex]
Simplify:
[tex]x+7 = \pm \sqrt{161}[/tex]
Subtract 7 from both sides we have;
[tex]x = -7 \pm \sqrt{161}[/tex]
Since, the sides cannot be in negative;
⇒[tex]x = -7+\sqrt{161}[/tex]
Length of patio = [tex]x+8 = -7+\sqrt{161}+8 = 1+\sqrt{161}[/tex] feet
Width of patio = [tex]x+6 = -7+\sqrt{161}+6 = -1+\sqrt{161}[/tex] feet
Therefore, the dimensions of the patio are:
[tex]1+\sqrt{161}[/tex] feet and [tex]-1+\sqrt{161}[/tex] feet
