Overview

Advanced algebra focuses on structure: factor clever expressions, transform systems, and use symmetry.

Key Ideas

• Completing the square and Vieta's formulas connect roots to coefficients.
• Symmetric systems often become simpler by letting $s=x+y$ and $p=xy$.
• For expressions like $x^2+y^2$, use $(x+y)^2-2xy$.

Worked Example

Solve $x+y=5$ and $x^2+y^2=13$.

Compute $2xy=(x+y)^2-(x^2+y^2)=25-13=12$, so $xy=6$. Then $t^2-5t+6=0$, giving $t=2$ or $3$. So $(x,y)$ is $(2,3)$ or $(3,2)$.

Practice Problems