Learn: Quadratic Equations

Welcome!Today we'll explore quadratic equations, a key topic in AQA A Level Mathematics. Understanding quadratics is important for solving many mathematical problems. Let's dive in step by step!

What is a Quadratic Equation?A quadratic equation is any equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. It is called 'quadratic' because the highest power of 'x' is 2.

Why Quadratics Are ImportantQuadratic equations are used to model many real-world scenarios, such as projectile motion or optimising areas. Solving them helps us find values of 'x' (roots) that make the equation true.

Standard Form of QuadraticsThe standard form is ax² + bx + c = 0. Each part has a role: 'a' affects the curve's width, 'b' affects its position, and 'c' is the y-intercept.

How to Solve Quadratic EquationsThere are three main methods: factorising, completing the square, and using the quadratic formula. We'll focus on each one briefly.

Factorising QuadraticsFactorising involves breaking the quadratic into two brackets, like (x + p)(x + q) = 0. Solve for 'x' by setting each bracket to zero: x = -p or x = -q.

Completing the SquareThis method rewrites the quadratic as a perfect square plus or minus a constant, like (x + p)² - q = 0. It helps when solving or sketching graphs.

The Quadratic FormulaIf factorising or completing the square isn't practical, use the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. It works for all quadratics.

Review Time!Great work! You've learned about quadratic equations, their forms, and methods to solve them. Let's review what we've covered through some questions.