Overview
This lecture covers key methods and concepts in quadratics, including factorising, using the quadratic formula, completing the square, the discriminant, interpreting quadratic models, and sketching quadratics.
Factorising Quadratics
- Factorising ax² + bx + c involves finding two numbers that multiply to ac and sum to b.
- When the coefficient of x² is not 1, factor out the common factor or use the “ac method.”
- To solve, set each bracket to zero and solve for x.
Quadratic Formula
- The quadratic formula: x = [-b ± √(b² - 4ac)] / 2a solves any quadratic equation.
- Always use brackets when squaring negative numbers to avoid errors.
- If b is negative, -b becomes positive in the formula.
Completing the Square
- To complete the square, the coefficient of x² must be 1; factorise if necessary.
- For ax² + bx + c, factor out a, complete the square on the inside, then expand back.
- The turning point (vertex) is at (opposite sign of value in the bracket, y-value at the end).
- For equations like 2(x - 2)² - 24 = 0, isolate the squared term, then solve for x.
The Discriminant
- The discriminant, Δ = b² - 4ac, determines the nature of roots:
- Δ > 0: Two real, distinct roots.
- Δ = 0: One real, repeated root.
- Δ < 0: No real roots (does not cross x-axis).
- The sign of the discriminant can be used to quickly determine if a quadratic has real roots.
Quadratic Models & Applications
- In physics problems (e.g., projectile motion), the constant term gives the initial value (like height).
- To find when the object hits the ground, solve the quadratic for the variable (e.g., set height = 0).
- Completed square form identifies the maximum or minimum value and its position (turning point).
Sketching Quadratics
- Y-intercept is given by the constant term.
- X-intercepts (roots) are found by solving the quadratic.
- The vertex/turning point is found from completed square form.
Key Terms & Definitions
- Quadratic equation — An equation of the form ax² + bx + c = 0.
- Quadratic formula — Solves quadratic equations: x = [-b ± √(b² - 4ac)] / 2a.
- Complete the square — Rewriting a quadratic in the form a(x + h)² + k.
- Discriminant — The part of the formula under the root: b² - 4ac; determines root nature.
- Turning point/vertex — The maximum or minimum point of a quadratic curve.
Action Items / Next Steps
- Practice factorising and solving quadratics using all methods.
- Review the quadratic formula and completing the square with negative coefficients.
- Complete any assigned homework on sketching and interpreting quadratic models.
- Watch linked revision videos for topics where extra help is needed.