SOLUTION: The length of rectangle is 3 cm more than 3 times the width and 1 cm less than the length of a diagonal. Find the length of the rectangle

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Question 156438: The length of rectangle is 3 cm more than 3 times the width and 1 cm less than the length of a
diagonal. Find the length of the rectangle
Answer by gonzo(654) (Show Source):
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width = w
length = 3w + 3 (3 times the width plus 3 centimeters)
diagonal = 3w + 4 (length is 1 cm less than diagonal which means diagonal is 1 cm more then length (3w + 3 is the length + 1 = 3w + 4 is the diagonal)).
solve for w using pythagorean theorem (length squared plus width squared = diagonal squared)since the diagonal is the hypotenuse of a right triangle formed by the length and the width and the diagonal.
equation becomes:
length (3w+3) squared plus width (w) squared = hypotenuse (3w+4) squared.
becomes (3w+3)^2 + w^2 = (3w+4)^2.
becomes 9w^2 + 18w + 9 + w^2 = 9w^2 + 24w + 16
becomes w^2 - 6w - 7 = 0
becomes (w-7) * (w+1) = 0
w = 7 or w = -1.
can't be -1 so w = 7
w = 7 then length = 3w + 3 = 24 and diagonal = 3w + 4 = 25.
width = 7
length = 24
diagonal = 25
check pythagorean theorem 7^2 + 24^2 = 25^2
becomes 49 plus 576 = 625
becomes 625 = 625.