Factoring the leading -1 from the first two terms, we have ...
... y = -(x² -6x) -6
Adding 9 inside parentheses to complete the square, then adding the opposite amount outside parentheses, we have ...
... y = -(x² -6x +9) -6 +9
... y = -(x -3)² +3
Answer:
A. 26 ft
Step-by-step explanation:
You might recognize that this triangle is a 5-12-13 triangle scaled up by a factor of 2. If so, you know immediately, that the hypotenuse is 2×13 = 26 feet in length.
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If you're not so lucky, you can write equations involving the lengths of the legs of the triangle. Let those be x and y, with x being the shorter leg. Then the Pythagorean relation tells you ...
x² + y² = (x+16)²
and the area relation tells you ...
120 = (1/2)xy
Using the second equation to write an expression for y, we have ...
y = 240/x
Substituting this into the first equation gives ...
x² + (240/x)² = (x+16)²
(240/x)² = 32x +256 . . . . subtract x²
1800 = x²(x +8) . . . . . . . . .multiply by x²/32
This is a cubic with one real root. It can be solved several ways. I like "graphing" as it is quick and easy. Trial and error also works. The factors of 1800 include 10²·18, so it is clear that x=10 is the solution.
The hypotenuse length is x+16, so is 26 feet.
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The attachment shows the graphical solution of the original two equations. This tells you the legs of the triangle are 10 and 24, so the hypotenuse will be 10+16 = 26. The 10-24-26 triangle is recognizable as a scaled version of the 5-12-13 triangle often used in math problems.
