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ALGEBRA

QUADRATICS

Expanding & Factorising 

An expression is a collection of algebraic terms. A quadratic expression is where the term with the highest power is a power of two. This section is all about quadratics and so first we look at expanding and simplifying quadratic expressions. 

Key Terms

  • Quadratic: an expression where the term with the highest power is an x² term. 

  • Expanding brackets: Removing the brackets from an expression. 

  • Factorising: Adding brackets into an expression. 

  • Monic: Where the coefficient of x² is one. 

  • Non-monic: Where the coefficient of x² isn't one. 

Solving Quadratic Equations (Factorising)

An equation is essentially two expressions connected with an equals sign. A quadratic equation can be rearranged to give an equation in the form ax²+bc+c=0. There are four main methods for solving quadratic equations: factorising, using the quadratic formula, completing the square and graphing. In this section, we look at the most simple method which is factorising. 

Prior knowledge

Key Terms

  • Zero product property: for two numbers to multiply to make zero, at least one of those numbers must be zero.

  • Equation: Two expressions connected with an equals sign. 

The Quadratic Formula

Not all quadratic equations factorise and so when we are faced with this situation, we need a method that we can rely on. The quadratic formula is a handy formula that will solve any quadratic equation. It can take a bit of time to memorise, but it is relatively simple to use. In this section, we will cover what it is, and how we can use it to solve any quadratic equation. 

Prior knowledge

Key Terms

  • Quadratic formula: a formula that can be used to solve any quadratic equation. 

  • General form of a quadratic equation: Quadratic equations written in the form ax²+bx+c=0.

Completing the Square

Completing the square is the process of finding the closest perfect square that fits into a quadratic expression. It can be used to find turning points of quadratic functions, and solve quadratic equations. 

Key Terms

  • Perfect square: a square number. 

  • Turning point: The coordinates of the point that quadratic graph changes direction.

Video Walkthrough 

Completing the Square 

Graphs of Quadratics 

Quadratic functions have a shape that looks similar to a happy or sad face, depending on whether the coefficient of x² is positive or negative. We call this shape a parabola, and can sometimes use the parabola to estimate the roots of quadratic equations. A quadratic graph will also have a turning point, which is the point where the graph changes direction. In this section, we will look at how to graph a quadratic function, and use our graph to identify some key properties. 

Prior knowledge

Key Terms

  • Parabola: The shape of a quadratic graph. 

  • Turning point: The coordinates of the point that a quadratic graph changes direction.

  • Roots: The "x-intercepts". Or, the points where the function is equal to zero. 

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