A square and a rectangle have the same area. The length of the rectangle is twelve more than twice times the side of the square. The width of the rectangle is eight less than the side of the square. Let x be the length of the side of the square. Using the information above, what is the value of x?

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MrRoyal MrRoyal · Answer 1 · 2020-11-11T02:05:56+01:00

Answer:

[tex]x = 12[/tex]

Step-by-step explanation:

Given

Shapes: Square and Rectangle

Square:

[tex]Length = x[/tex]

Rectangle:

[tex]Length = 12 + 2x[/tex]

[tex]Width = x- 8[/tex]

Required

Find x

First, we calculate the area of the square.

[tex]Area = x * x[/tex]

[tex]Area = x^2[/tex]

Next, we calculate the area of the rectangle

[tex]Area = (12 + 2x) * (x - 8)[/tex]

[tex]Area = 12x -96 + 2x^2 - 16x[/tex]

[tex]Area = 2x^2 - 16x + 12x -96[/tex]

[tex]Area = 2x^2 -4x -96[/tex]

Since both areas are equal, we have:

[tex]x^2 = 2x^2 -4x -96[/tex]

Equate to 0

[tex]2x^2 -x^2 -4x -96= 0[/tex]

[tex]x^2 -4x -96= 0[/tex]

Expand:

[tex]x^2 - 12x + 8x -96 = 0[/tex]

[tex]x(x - 12) + 8(x - 12) = 0[/tex]

[tex](x + 8)(x - 12) = 0[/tex]

[tex]x + 8 = 0[/tex] or [tex]x - 12 = 0[/tex]

[tex]x = -8[/tex] or [tex]x = 12[/tex]

But x can't be negative.

Hence:

[tex]x = 12[/tex]