Learn: Quadratics

Welcome!Today we'll learn about quadratics. Quadratics are an important part of algebra and involve expressions where the highest power of the variable is 2. Let's explore step by step!

What are Quadratic Equations?A quadratic equation is a type of equation in the form ax² + bx + c = 0, where a, b, and c are numbers, and x is the variable. It's called 'quadratic' because it involves x² (which means x multiplied by itself).

Solutions to Quadratic EquationsQuadratic equations can have up to two solutions. These solutions are the values of x that make the equation equal to 0. We can find them using methods like factorisation, the quadratic formula, or completing the square.

Factorising QuadraticsFactorisation involves breaking a quadratic equation like x² + 5x + 6 into two brackets: (x + 2)(x + 3). When expanded, it equals the original equation. This method works when the quadratic can be expressed as a product of two linear factors.

The Quadratic FormulaThe quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It works for any quadratic equation and gives the solutions directly. The term under the square root, b² - 4ac, is called the discriminant.

Completing the SquareCompleting the square rewrites a quadratic equation into the form (x + p)² + q = 0. This method is helpful for finding solutions and understanding the graph of the equation.

Graphing QuadraticsThe graph of a quadratic equation is a parabola. If the coefficient of x² (a) is positive, the parabola opens upwards, and if negative, it opens downwards. The solutions of the equation are where the graph crosses the x-axis.

Review Time!Well done! You've learned how to solve quadratic equations using factorisation, the quadratic formula, and completing the square. You've also explored the graph of quadratics. Let's test your understanding!