7.) What are the possible expressions for the dimensions of the cuboid whose volume=18x square+15xy-18y square
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I think you're probably being asked to factor these expressions fully, but really, there are an infinite number of possible expressions for the dimensions. For example:
(3x2 - 12x), (1/x), and (x) are possible dimensions of the first cuboid
(3x2 - 12x), (x4-19), and (1/(x4-19)) are possible dimensions as well
But let's ignore that and just fully factor each expression for volume.
1.
3x2 - 12x
3(x2 - 4x)
3x(x - 4)
There are only three factors, so there is only one combination of possible dimensions:
3, x, and (x - 4)
2.
12ky2 + 8ky - 20k
4(3ky2 + 2ky - 5k)
4k(3y2 + 2y - 5)
4k(3y + 5)(y - 1)
Now we've got four factors, so there are several combinations of possible dimensions:
4, k, and (3y2 + 2y - 5)
4, (3y + 5), and (ky - k)
k, (3y + 5), and (4y - 4)
4, (y - 1), and (3ky + 5k)
k, (y - 1), and (12y + 20)
(3y + 5), (y - 1), and 4k
18x^2+15xy-18y
^2= equation:
Simplifying
18x2 + 15xy + -18y2 = 0
Reorder the terms:
15xy + 18x2 + -18y2 = 0
Solving
15xy + 18x2 + -18y2 = 0
Solving for variable 'x'.
Factor out the Greatest Common Factor (GCF), '3'.
3(5xy + 6x2 + -6y2) = 0
Factor a trinomial.
3((3x + -2y)(2x + 3y)) = 0
Ignore the factor 3.
Subproblem 1
Set the factor '(3x + -2y)' equal to zero and attempt to solve:
Simplifying
3x + -2y = 0
Solving
3x + -2y = 0
Move all terms containing x to the left, all other terms to the right.
Add '2y' to each side of the equation.
3x + -2y + 2y = 0 + 2y
Combine like terms: -2y + 2y = 0
3x + 0 = 0 + 2y
3x = 0 + 2y
Remove the zero:
3x = 2y
Divide each side by '3'.
x = 0.6666666667y
Simplifying
x = 0.6666666667y
Subproblem 2
Set the factor '(2x + 3y)' equal to zero and attempt to solve:
Simplifying
2x + 3y = 0
Solving
2x + 3y = 0
Move all terms containing x to the left, all other terms to the right.
Add '-3y' to each side of the equation.
2x + 3y + -3y = 0 + -3y
Combine like terms: 3y + -3y = 0
2x + 0 = 0 + -3y
2x = 0 + -3y
Remove the zero:
2x = -3y
Divide each side by '2'.
x = -1.5y
Simplifying
x = -1.5y
Solution
x = {0.6666666667y, -1.5y}
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