find the value of x^3+y^3-12xy+64 when x+y= -4
Nothing is more fun that cubic equations, in school, most are trick questions, such as this. In real life, they are challenging.
x^3+y^3-12xy +64=A
(x+y)(x^2-xy+y^2) -12xy +64=A
now divide through by -4
x^2-xy+y^2+3xy+64=A/4
x^2+2xy+y^2 =A/4-64
(x+y)^2=A/4-64
16=A/4-64
A=320 which answers the question.
Better check the ± signs
To find the value of x^3+y^3-12xy+64 when x+y=-4, we can use the given equation to express one variable in terms of the other variable.
Given: x+y=-4
Let's solve this equation for x:
x = -4 - y
Now we substitute this expression for x into the expression x^3+y^3-12xy+64:
(-4 - y)^3 + y^3 - 12(-4 - y)y + 64
Next, we simplify and evaluate the expression:
(-4 - y)^3 = (-64 - 48y -12y^2 - y^3)
y^3 - 12xy = y^3 - 12(-4 - y)y
Expanding (-4 - y)^3 gives us:
-64 - 48y - 12y^2 - y^3 + y^3 - 12xy + 64
Simplifying further:
(-64 - 48y - 12y^2 - y^3) + y^3 - 12(-4 - y)y + 64
Combining like terms:
-48y - 12y^2 - 12(-4 - y)y
Expanding:
-48y - 12y^2 + 48y + 12y^2
The -48y and +48y terms cancel each other out, as do the -12y^2 and +12y^2 terms, leaving us with:
-12y^2
Therefore, the value of x^3+y^3-12xy+64 when x+y=-4 is -12y^2.
In summary, the value of x^3+y^3-12xy+64 when x+y=-4 is -12y^2.