if x=2y+6 find the value of x cube-8y cube -36xy-216
x^3-8y^3-36xy-216
(x-2y)(x^2+2xy-4y^2)-72y^2-108-216
(6)(4y^2-24y+4y^2)-72y^2-324
check those lines carefully, I did them in my head. If correct, then combine terms...
[x^3-8y^3] -36xy-216
= (x - 2y)(x^2 + 2xy + 4y^2) - 36(xy + 6)
= 6((2y+6)^2 + 2y(2y+6) + 4y^2) - 36(y(2y+6) + 6)
= 6( 4y^2 + 24y + 36 + 4y^2 + 12y + 4y^2) - 36(2y^2 + 6y + 6)
= 24y^2 + 144y + 216 + 24y^2 + 72y + 24y^2 - 72y^2 - 216y - 216
= 0
check:
let y = 1, then x = 8
x^3 - 8y^3 - 36xy - 216
= 512 - 8 - 288 - 216 = 0
from my answer:
= 0
my answer is reasonable
To find the value of x^3 - 8y^3 - 36xy - 216, we first need to simplify the expression by substituting the value of x from the given equation.
Given: x = 2y + 6
Substituting x = 2y + 6 into the expression, we get:
(2y + 6)^3 - 8y^3 - 36xy - 216
To simplify this, we need to expand (2y + 6)^3 using the binomial expansion formula:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
In our case, a = 2y and b = 6, so substituting into the formula, we have:
(2y)^3 + 3(2y)^2*6 + 3(2y)*(6^2) + 6^3 - 8y^3 - 36xy - 216
Simplifying further:
8y^3 + 3(4y^2)*6 + 3(2y)*(36) + 216 - 8y^3 - 36xy - 216
Notice that the terms -216 and +216 cancel out, simplifying the expression to:
8y^3 + 72y + 108 - 8y^3 - 36xy
Now, we can combine like terms:
(8y^3 - 8y^3) + 72y - 36xy + 108
The terms with the same exponent but different signs cancel out:
0 + 72y - 36xy + 108
Finally, we can factor out a common factor:
72y - 36xy + 108 = 36(2y - x + 3)
So, the simplified expression is 36(2y - x + 3).