Overview
This lecture explains how to complete the square for quadratic equations, including practical steps, when to use the technique, and its importance for graphs and proofs.
What is Completing the Square?
- Completing the square is a method to solve quadratic equations and rewrite them in vertex/turning point form.
- It transforms a quadratic y = ax² + bx + c into y = a(x + p)² + q.
- This form helps to identify the turning point coordinates of the quadratic graph.
How to Complete the Square
- If the coefficient a = 1, p = b/2 and q = c - (b/2)².
- If a ≠1, first factor a out from the x² and x terms, then complete the square as above.
- The method allows you to rewrite the quadratic to show key features such as turning points.
Uses of Completing the Square
- It finds the turning point (vertex) of a quadratic graph easily.
- It helps create a quadratic equation when given the turning point.
- The technique can be used to show that a squared term is always ≥ 0, useful for proofs and showing results.
Exam Tips
- Some questions directly ask for completing the square; sometimes you need to identify when to use it, especially for finding turning points.
- Remember the standard form to write your answer unless told otherwise.
Key Terms & Definitions
- Quadratic equation — an equation of the form y = ax² + bx + c.
- Completing the square — rewriting a quadratic as y = a(x + p)² + q.
- Turning point (vertex) — the maximum or minimum point of a quadratic graph.
- Coefficient — the number multiplying a variable (e.g., 'a' in ax²).
Action Items / Next Steps
- Practice rewriting different quadratics by completing the square for both a = 1 and a ≠1.
- Try using the completed square form to find turning points of quadratics.
- Attempt exam questions where completing the square is needed for proofs or graph analysis.