9709 · 1.1
Quadratics
A quadratic function creates a U-shaped curve called a parabola. By understanding its equation, we can unlock all its key features, like where it crosses the axes and its highest or lowest point.
Need to know
What you need to know
- **Factorisation:** If the quadratic expression can be written as a product of two linear factors, $(px+q)(rx+s)=0$, then the solutions are $x = -q/p$ and $x = -s/r$. This is the quickest method when it's possible.
- **Completing the Square:** This method rewrites the equation to isolate $x$. It's a reliable method that also helps in finding the vertex of the parabola.
- **The Quadratic Formula:** This formula works for any quadratic equation and is essential when factorisation is not straightforward. It is derived from the method of completing the square.
Explanation
The Parabola's Secrets
- A quadratic equation y = ax² + bx + c has a parabolic graph (a ≠ 0).
- Completing the square reveals the vertex form y = a(x − h)² + k.
- The discriminant b² − 4ac determines the nature of roots.
- Factorise or use the formula to solve ax² + bx + c = 0 on Paper 1.