The house of Mr. Hashim has a rectangular table center. He wants to put a mantle with a length of 2 longer than its width. The area is 24m2. What will be the length and width of the mantle

Problem: The house of Mr. Hashim has a rectangular table center. He wants to put a mantle with a length of 2 longer than its width. The area is 24m². What will be the length and width of the mantle?

Solution: Represent l and w as the length and width of the rectangle respectively. Make equations of the given statement.

[tex] \begin{cases}l = w + 2& \green{(eq. \: 1)} \\ 24 = l \cdot w& \green{(eq. \: 2)}\end{cases}[/tex]

- Substitute l from the first equation to the second equation in terms of w.

[tex] \begin{cases}l = w + 2 \\ 24 = (w + 2) \cdot w\end{cases}[/tex]

[tex] \begin{cases}l = w + 2 \\ 24 = {w}^{2} + 2w \end{cases}[/tex]

[tex] \begin{cases}l = w + 2 \\ {w}^{2} + 2w - 24 = 0 \end{cases}[/tex]

- Solve the quadratic equation from the second equation by factoring.

[tex] {w}^{2} + 2w - 24 = 0[/tex]

[tex](w - 4)(w + 6) = 0[/tex]

[tex]w - 4 = 0 \: \: \: \: ; \: \: \: \: w + 6 = 0[/tex]

[tex]w = 4 \: \: \: \: ; \: \: \: \: w = \text - 2[/tex]

[tex] \begin{cases}l = w + 2 \\w = 4\end{cases}[/tex]

- The width of the mantle is 4 m, substitute it to the first equation to find the length of the mantle.

[tex] \begin{cases}l = 4 + 2 \\w = 4\end{cases}[/tex]

[tex] \begin{cases}l = 6 \\w = 4\end{cases}[/tex]

- Therefore, the dimensions of the mantle are:

[tex] \large \rm Length = \boxed{ \rm \green{ \: 6 \: meters \: }}[/tex]

[tex] \large \rm Width = \boxed{ \rm \green{ \: 4 \: meters \: }}[/tex]

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