Problem
ALG-B1-M01-P004 Symmetric Sum
#4
★☆☆☆☆ Level 1 of 5
Let \(x+y=7\), \(xy=10\). Find \(x^2+y^2\) and \(x^3+y^3\) without finding \(x\) and \(y\) separately.
Use \(x^2+y^2=(x+y)^2-2xy\), then the sum of cubes formula.
We have \(x^2+y^2=7^2-2\cdot10=29\).
Also \(x^3+y^3=(x+y)(x^2-xy+y^2)=7(29-10)=133\).
The goal is to see symmetry, not solve a quadratic first.