X3+ 8y3=?
X+2y=8
XY=2
x^3 + 8y^3 is the sum of two cubes (x^2 + (2y)^3), sp
= (x+2y)(x^2-2xy+4y^2)
= 8(x^2-4+4y^2)
Now, what can we do with that?
(x+2y)^2 = x^2 + 4xy + 4y^2
8^2 = 4*3 + x^2+4y^2
x^2+4y^2 = 64-12 = 52
so, now we know that
x^3 + 8y^3 = 8(x^2-4+4y^2) = 8(52-4) = 384
oops. A typo. Should be
8^2 = 4*2 + x^2+4y^2
x^2 + 4y^2 = 56
x^3 + 8y^3 = 8(x^2-4+4y^2) = 8(56-4) = 416
or,
x^3 + 8y^3 = (x+2y)(x^2-2xy+4y^2)
= (x+2y)((x^2+4xy+4y^2) - 6xy)
= (x+2y)((x+2y)^2 - 6xy)
= 8(64-12) = 416
as above
Given:
Eq1: x^3 + 8y^3 = ?
Eq2: x + 2y = 8.
Eq3: xy = 2.
xy = 2.
Y = 2/x.
In Eq2, replace Y with 2/x:
x + 2*2/x = 8.
x + 4/x = 8,
Multiply both sides by X:
x^2 + 4 = 8x,
x^2 - 8x + 4 = 0.
Use Quadratic Formula to find X:
X = (-B +- sqrt(B^2-4AC))/2A.
X = (8 +- sqrt(48))/2 = 7.5, and 0.54.
In Eq3, replace X with 7.5:
7.5y = 2.
Y = 0.27.
In Eq1, replace X with 7.5 and replace Y with 0.27.
To find the value of x^3 + 8y^3, we need to solve the given equations.
The first equation is x + 2y = 8. We can solve for x by isolating the variable:
x = 8 - 2y
Next, we have the equation XY = 2. Substituting x from the first equation:
(8 - 2y) * y = 2
Expanding the equation:
8y - 2y^2 = 2
Rearranging the equation:
2y^2 - 8y + 2 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -8, and c = 2. Substituting these values into the quadratic formula:
y = (-(-8) ± √((-8)^2 - 4 * 2 * 2)) / (2 * 2)
Simplifying the equation further:
y = (8 ± √(64 - 16)) / 4
y = (8 ± √48) / 4
y = (8 ± 4√3) / 4
Now, we can find the values of y:
Case 1: y = (8 + 4√3) / 4
Simplifying:
y = 2 + √3
Case 2: y = (8 - 4√3) / 4
Simplifying:
y = 2 - √3
Now that we have the values of y, we can substitute them back into the first equation to find the corresponding values of x. Let's calculate them:
Case 1: y = 2 + √3
Substituting into x + 2y = 8:
x + 2(2 + √3) = 8
x + 4 + 2√3 = 8
x = 8 - 4 - 2√3
x = 4 - 2√3
Case 2: y = 2 - √3
Substituting into x + 2y = 8:
x + 2(2 - √3) = 8
x + 4 - 2√3 = 8
x = 8 - 4 + 2√3
x = 4 + 2√3
Now, we can substitute the found x and y values into the expression x^3 + 8y^3 to get the final answer:
Case 1: x^3 + 8y^3 = (4 - 2√3)^3 + 8(2 + √3)^3
Expanding and simplifying further:
x^3 + 8y^3 = 64 - 96√3 + 72 - 36√3 + 27 - 18√3 + 8(8 + 12√3 + 6√3 + 2√3)
x^3 + 8y^3 = 163 - 138√3 + 32√3
x^3 + 8y^3 = 163 - 106√3
Case 2: x^3 + 8y^3 = (4 + 2√3)^3 + 8(2 - √3)^3
Expanding and simplifying further:
x^3 + 8y^3 = 64 + 96√3 + 72 + 36√3 + 27 - 18√3 + 8(8 - 12√3 + 6√3 - 2√3)
x^3 + 8y^3 = 163 + 102√3
So, the two solutions are:
Solution 1: x^3 + 8y^3 = 163 - 106√3
Solution 2: x^3 + 8y^3 = 163 + 102√3