the sum of two numbers is 1 and the sum of their square is 13. What are the sum of their cubes?

Question

Is the answer 35?

Answer 1

call the first number x, the second number y. since their sum is 1,

x+y=1

since the sum of their squares is 13,

x²+y²=13

 

since x+y=1

y=1-x

Replace that in to the second equation.

x²+(1-x)²=13    foil it out

x²+1-2x+x²=13   so

2x²-2x+1=13    subtract 13 from both sides

2x^x-2x-12=0      factor out a 2

2(x²-x-6)=0

2(x-3)(x+2)=0

 

so x=3, -2

 

since y=1-x,

if x=3, then y=-2

if x=-2, then y=3

 

so check. does 3+(-2)=1? Yes

(3)²+(-2)²=13? Yes

 

so check. does (-2)+3=1? Yes

(-2)²+(3)²=13? Yes.

 

So for x=3 and y=-2,

(3)³+(-2)³=35.

 

So for x=-2 and y=3,
(-2)³+(3)³=35.

 

Yes, you are correct.