1. Let's start by stating the problem: understanding quadratics as taught in Year 9 Pearson Maths. 2. A quadratic equation is any equation that can be written in the form $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are constants and $a \neq 0$. 3. The graph of a quadratic equation is a parabola, which can open upwards if $a > 0$ or downwards if $a < 0$. 4. Key features of a quadratic include the vertex (the highest or lowest point), the axis of symmetry (a vertical line through the vertex), and the roots or x-intercepts (where the graph crosses the x-axis). 5. The vertex can be found using the formula for the x-coordinate: $$x = -\frac{b}{2a}$$. Substitute this back into the equation to find the y-coordinate. 6. The roots can be found by solving the quadratic equation using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 7. The discriminant $$\Delta = b^2 - 4ac$$ tells us about the nature of the roots: if $$\Delta > 0$$, two real roots; if $$\Delta = 0$$, one real root; if $$\Delta < 0$$, no real roots. 8. Quadratics can also be factorised if possible, for example, $$y = x^2 + 5x + 6$$ factors to $$(x + 2)(x + 3)$$. 9. Completing the square is another method to rewrite quadratics in vertex form: $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. 10. Understanding these concepts helps solve problems involving projectile motion, area, and optimization in Year 9 maths. Final answer: Quadratics are equations of the form $$y = ax^2 + bx + c$$ with key features including vertex, axis of symmetry, and roots, which can be found using formulas and methods like factorisation and completing the square.