Overview
This lecture covers five main methods for solving quadratic equations, explains how each method works, and suggests when to use each one.
Methods for Solving Quadratic Equations
1. Graphing Method
- Set the quadratic equation equal to zero and rewrite as y = equation.
- Graph the parabola and identify x-intercepts (where the graph crosses the x-axis) as solutions.
- Use this method mainly if you have a graphing calculator or a pre-drawn graph.
- Not ideal if solutions are not neat integers or graphing is impractical.
2. Quadratic Formula
- Use the formula: x = [-b ± √(b² – 4ac)] / (2a) for equations in ax² + bx + c = 0 form.
- Efficient when the equation is hard to factor or graph.
- Provides exact or approximate solutions using a calculator.
3. Factoring
- Suitable when the quadratic can be easily factored.
- Find two numbers that multiply to the constant term and add to the linear coefficient.
- Set each factor equal to zero to find the solutions.
4. Completing the Square
- Move constants to the other side, then add (b/2)² to both sides to create a perfect square trinomial.
- Solve by taking the square root of both sides, giving two solutions (plus and minus).
- Best used when the coefficient of x is even and the leading coefficient is one.
5. Taking Square Roots
- Isolate the squared term (e.g., ax² = c), then take the square root of both sides.
- Use this method when there is no x (linear) term, just x².
- Don’t forget the plus/minus when taking square roots.
Key Terms & Definitions
- Quadratic Equation — An equation in the form ax² + bx + c = 0.
- Vertex — The turning point of the parabola, found at x = –b/(2a).
- Axis of Symmetry — The line x = –b/(2a) dividing the parabola into two mirror images.
- Quadratic Formula — Formula to find roots: x = [-b ± √(b² – 4ac)] / (2a).
- Factoring — Rewriting a quadratic as (x + p)(x + q) = 0 to find solutions.
- Completing the Square — Transforming ax² + bx + c into a perfect square trinomial for solving.
- Square Root Method — Solving equations like x² = k by taking square roots.
Action Items / Next Steps
- Practice using each method on at least one example quadratic equation.
- Review your notes on the quadratic formula and factoring techniques.
- Identify which solving method works best for different types of quadratic equations.