1. Nonlinear Equations in One Variable

• Quadratic Equations: Solve by factoring, completing the square, or using the quadratic formula.
• Zero-product property: \( ab = 0 \Rightarrow a = 0 \text{ or } b = 0 \)
• Discriminant: \( b^2 - 4ac \) tells number of real solutions
• Absolute Value Equations: Solve \( |x| = a \Rightarrow x = a \text{ or } x = -a \)
• Square Root Equations: Solve by squaring both sides (watch for extraneous solutions)

2. Systems of Equations Involving Nonlinear Functions

• Linear-Quadratic Systems: Solve for points of intersection algebraically or graphically
• Quadratic-Quadratic Systems: Set two quadratic expressions equal and solve
• Algebraic Techniques: Use substitution or elimination with nonlinear terms
• Graphical Interpretation: Analyze curves to determine number of intersections (0, 1, or 2)

3. Application-Based Problems

• Physics and Motion: Model with velocity, acceleration, or height equations
• Business: Use cost/revenue functions involving quadratics
• Optimization: Apply quadratic maximum/minimum techniques in real-world word problems

Summary

These problems go beyond simple algebra, requiring you to:

• Solve and interpret quadratic, absolute value, and root-based equations
• Use the discriminant to find number of real solutions
• Understand how lines and curves intersect in nonlinear systems
• Apply algebraic strategies and graphical insights to real-world applications

Key Skills

• Use the discriminant to classify solutions: \( b^2 - 4ac \)
• Factor and complete the square
• Express one variable in terms of another
• Find positive and negative solutions in absolute value problems
• Graph and solve nonlinear systems algebraically and visually