Learn: Quadratics

Welcome!I've looked at your growth areas and created this lesson to help strengthen your understanding of quadratics. We'll focus on key concepts like solving quadratic equations, graphing, and understanding their properties.

What are Quadratics?A quadratic equation is an equation in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. Quadratics are important because they model many real-world scenarios, like projectile motion or business problems involving profit.

Key Features of QuadraticsQuadratic equations have a U-shaped graph called a parabola. The highest or lowest point on the graph is called the vertex, and the graph may cross the x-axis at up to two points called roots or solutions.

Solving Quadratic EquationsThere are three main methods for solving quadratic equations: factorising, using the quadratic formula, or completing the square. Factorising works when the equation can be expressed as a product of two brackets, like (x + 2)(x - 3) = 0.

Factorising QuadraticsTo factorise a quadratic like x² + 5x + 6 = 0, find two numbers that multiply to the constant term (6) and add to the coefficient of x (5). Here, 2 and 3 work because 2 × 3 = 6 and 2 + 3 = 5. This gives (x + 2)(x + 3) = 0.

The Quadratic FormulaThe quadratic formula is used when factorising is not possible. It is x = [-b ± √(b² - 4ac)] / 2a. This formula works for any quadratic equation, and it uses the coefficients a, b, and c from ax² + bx + c = 0.

Completing the SquareCompleting the square rewrites a quadratic equation in the form (x + p)² + q = 0. For example, x² + 6x + 8 can be rewritten as (x + 3)² - 1 = 0. It’s helpful for finding the vertex and solving equations.

Graphing QuadraticsThe graph of a quadratic equation shows its roots, vertex, and whether it opens upwards (positive a) or downwards (negative a). Drawing the graph helps visualise solutions.

Review Time!Great work! You've learned about solving quadratics, graphing them, and understanding their properties. Let's test your understanding with a few questions.