Quadratic Expressions
Year 9 📊 Algebra Expand and factorise quadratics, difference of two squares.
📐 Quadratic Expressions and Parabolas
A quadratic expression has the form $ax^2 + bx + c$ where $a \neq 0$. Its graph is a curved shape called a parabola.
If $a > 0$ the parabola opens upward (U-shape) and has a minimum. If $a < 0$ it opens downward and has a maximum.
🔍 Factorising Quadratics ($a > 1$)
When the coefficient of $x^2$ is greater than 1, use the AC method.
- Calculate $ac$ (multiply the $x^2$ coefficient by the constant).
- Find two numbers that multiply to $ac$ and add to $b$.
- Split the middle term using these numbers.
- Factorise by grouping.
Example: Factorise $2x^2 + 7x + 3$. Here $ac = 6$. Numbers: 6 and 1 (multiply to 6, add to 7). → $2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)$
📊 Completing the Square
Completing the square rewrites a quadratic in the form $(x+p)^2 + q$, revealing the vertex of the parabola.
⚡ Method
$$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$Example: $x^2 + 6x + 2 = (x+3)^2 - 9 + 2 = (x+3)^2 - 7$. The vertex (minimum point) is at $(-3, -7)$.
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Interactive Demonstration — Quadratic Expressions
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📐 Quadratic Explorer
Factorise quadratic expressions ax² + bx + c.