We need to account for both x values on either side of the length, and width.
Thus, the length becomes 10 + x + x = 10 + 2x
and the width becomes 5 + x + x = 5 + 2x
For the second question, I'm assuming we don't account for the area that is covered by the garden.
Then we can say that the path is measured by: (5 + 2x)(10 + 2x) - 50, which is the area of the garden itself.
(5 + 2x)(10 + 2x) - 50 = 54
Expanding the brackets:
[tex]50 + 10x + 20x + 4x^{2} - 50 = 54[/tex]
[tex]4x^{2} + 30x = 54[/tex]
[tex]2x^{2} + 15x - 27 = 0[/tex]
[tex]2x^{2} - 3x + 18x - 27 = 0[/tex]
[tex]x(2x - 3) + 9(2x - 3) = 0[/tex]
[tex](x + 9)(2x - 3) = 0[/tex]
x = -9, or x = 3/2
Since x > 0, then x ≠ -9
Thus, the only x-value we can take is x = 3/2
