Hello, fladam!

A square and a rectangle have equal perimeters.

The length of the rectangle is 11cm.

The area of the square is 4cm² larger than the area of the rectangle.

Find the length of a side on the square. Code:

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| | | | x
w | 11w | | x² |
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11 x

The perimeter of the rectangle is: $\displaystyle 2w + 22$

The perimeter of the square is: $\displaystyle 4x$

The perimeters are equal: .$\displaystyle 2w + 22 \:=\:4x \quad\Rightarrow\quad w \:=\:2x-11\;\;{\color{blue}[1]}$

The area of the rectangle is: $\displaystyle 11w$

The area of the square is: $\displaystyle x^2$

The area of the square is 4 more than that of the rectangle: .$\displaystyle x^2 \:=\:11w + 4\;\;{\color{blue}[2]}$

Substitute [1] into [2]: .$\displaystyle x^2 \:=\:11(2x-11) +4\quad\Rightarrow\quad x^2 - 22x + 117 \:=\:0$

Factor: .$\displaystyle (x-9)(x-13) \:=\:0$

And we have two solutions: .$\displaystyle x \:=\:9,\:13$

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**Check**

If $\displaystyle x = 9$, substitute into [1]: .$\displaystyle w \:=\:2(9) - 11 \:=\:7$

The rectangle is 7 × 11.

Its perimeter is 36 cm.

Its area is 77 cm².

The square is 9 × 9.

Its perimeter is 36 cm.

Its area is: 81 cm².

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If $\displaystyle x = 13$, then: .$\displaystyle w \:=\:2(13)-11 \:=\:15$

The rectangle is: 15 × 11.

Its perrimer is 52 cm.

Its area is: 165 cm².

The square is: 13 × 13.

Its perimeter is: 52 cm.

Its area is: 169 cm².

Both solutions check out!