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Question

x+y=1
x^2+y^2=4
what is X^3+Y^3 equal to plz show your works

is it 4-xy

Answer 1

11/2.
x+y=1 => x^2+y^2+2*xy = 1 => xy = -3/2
X^3+Y^3 = (x+y)(x^2+y^2-xy)=11/2

Answer 2

Solve the system of equations. If x+y=1, then y=1-x.
Thus,
x^2+y^2 = 4
x^2 + (1-x)^2 = 4
x^2 + 1 - 2x +x^2 = 4
2x^2 - 2x - 3 =0
From the quadratic formula, x=(1 + sqrt(7))/2 and (1 - sqrt(7))/2.

Substituting back in,
x^3 + y^3 = x^3 + (1-x)^3 = 11/2.

Note that either value of x will yield the same result.

Answer 3

y = 1 - x
x² + (1 - x)² = 4
x² + 1 - 2x + x² = 4
2x² - 2x - 3 = 0
x = 1 +/- √(4 + 24) / 2
x = 1 +/- √7
y = +/- √7

{x,y} = {1 + √7, -√7} or {1 - √7, √7}

x³ + y³ = (1 + √7)³ - 7√7 = 22 + 3√7
or
x³ + y³ = (1 - √7)³ + 7√7 = 22 - 3√7

So, the answer is 22 +/- 3√7

Answer 4

yes.

x^3 + y^3 = (x+y)(x² - xy + y²) = (1)(4 - xy)