Question

If x + y = 1 and  x3 + y3 = 19, then find the value of x2 + y2 .

Answer 1

x + y = 1
Squaring on both sides,
x²+y²+2xy = 1
x²+y² = 1 - 2xy    ....... (1)

x³+y³ = 19
(x+y)(x²+y²-xy) =19
x²+y²-xy =19     [Since x + y = 1]
x²+y² =19 + xy       .................(2)

From the equations (1) and (2)

1 - 2xy = 19 + xy

3xy = -18

xy = -6

Now, substitute the value of xy in equation (2)
x²+y² =19 + (-6) = 13