Algebra!

Join us in this step-by-step guide through the wonderful world of x's and y's, equations and operations, as we uncover the mysteries of Algebra.

Algebraic Terminology

Let's begin with the basic terminology.

Variable / Pronumeral

These are the x's and y's, something which we don't know the value of yet. The goal of algebra is generally to solve for these variables.

Coefficient

The number which is out the front of a variable, i.e. the number which is multiplied by the variable.

Constant

The number which is on its own, and doesn't impact the variable.

Term

Any combination of a coefficient and a variable.

The expression 3x + 2 labelled: 3 is the coefficient, x is the variable, 2 is the constant — In 3x + 2: 3 is the coefficient, x is the variable and 2 is the constant.

"If I had a dollar for every time Algebra has helped me, I'd have x dollars."

Algebra: Basic Operations

Just like how we can use operations on numbers, we can add, subtract, multiply, and divide variables as well.

Four examples of basic operations on variables: x + x = 2x, 3a − a = 2a, y × y = y squared, 4b ÷ b = 4 — Basic operations: x + x = 2x; 3a − a = 2a; y × y = y²; 4b ÷ b = 4.

Collecting Like Terms

To make our expressions smaller, we can sometimes collect like terms. These are terms which only differ by the coefficient.

Solving Algebraic Equations

We can now start to solve equations with the rules we've just learnt. The point of these equations is to find the value of the variable.

Method of solving equations

Move the constants
Move the constant terms to the right-hand side of the equation.
Divide by the coefficient
Divide both sides by the coefficient to leave the variable by itself.

Worked example: 1) 4x + 5 = 21; 2) 4x = 16 (subtract 5 from both sides); 3) x = 4 (divide by 4) — Worked example: 4x + 5 = 21 → 4x = 16 → x = 4.

"Finding a treasure is like working on algebraic equations, all you have to do is find the X."

Introduction to the Quadratic

The next step of our journey into the world of algebra is the quadratic, which is something you'll see over and over in your high school maths classes. A quadratic is an expression which contains an x² (x squared) term with no higher powers.

All quadratics have the following form, where a and b are both coefficients, and c is a constant.

The general form of a quadratic: a x squared plus b x plus c — The general form of a quadratic: ax² + bx + c.

Some Examples of Quadratics

Not all quadratics look the same. The only real identifier of a quadratic is that the highest power of x is 2.

x² + 5x + 4

This is very much a quadratic, it contains an x squared term.

x² – 9

This is also a quadratic, just with the b coefficient equal to 0.

x²

This example is still a quadratic, even though it only contains one term. The only difference here is both the b coefficient and c constant are equal to 0.

(x+1)²

This looks different from the others since we can't see any obvious x² term. However, as we'll discuss further on in this resource, this is still a quadratic expression.

Want to know what a quadratic looks like? Visit our free Graphing Resource to find out!

Expanding and Factorising Brackets

In order to solve quadratics, we need to know how to expand and factorise brackets.

If we have two brackets, each containing an x term, we may want to expand these brackets into a single quadratic. To expand, follow these 4 steps.

Expanding (x+a)(x+b) in four steps with arrows: 1) x times x gives x squared, 2) x times b gives ax, 3) a times x gives bx, 4) a times b gives ab, so (x+a)(x+b) = x squared + ax + bx + ab — Expanding brackets: (x+a)(x+b) = x² + ax + bx + ab.

To reverse this process is called factorising. Let's consider the example below, 3x² + 5x + 2.

Factorising 3x squared + 5x + 2: find two numbers that add to 5 and multiply to 6; rewrite as 3x squared + 3x + 2x + 2; group as 3x(x+1) + 2(x+1); factorise to (3x+2)(x+1) — Factorising 3x² + 5x + 2 = (3x + 2)(x + 1).

Multiply the a coefficient by the constant c (in our case 3 × 2).
Find two numbers which add to the b coefficient (5) and multiply to our product ac (6). In our case, this would be 3 and 2.
Split the b term up into a sum containing these two numbers.
Take out any common terms that appear. In our case, the first two terms both contain a 3x and the second two terms both contain a 2.
Join these two terms as shown to give a factorised form.

Not all quadratics can be factorised!

Sometimes you'll come across a quadratic that no matter how hard you try, you can't seem to factorise it. Don't stress, it's probably because it can't be done!

Special Quadratics

Perfect Squares and Difference of Two Squares

The perfect square and difference of two squares are special because they can be factorised easily without going through the tedious process shown above.

Perfect square: (a+b) squared = a squared + 2ab + b squared. Difference of two squares: (a+b)(a−b) = a squared − b squared — Perfect square: (a+b)² = a² + 2ab + b². Difference of two squares: (a+b)(a−b) = a² − b².

Solving Quadratic Equations

Now that we know what a quadratic is and how to factorise, we can start to solve them.

Shift
Shift everything to one side of the equation to give a quadratic on the left-hand side and 0 on the right-hand side.
Factorise
Factorise the quadratic (if it is impossible to factorise, then we say that this equation has no solutions).
Solve
Solve the equation by determining what values of x make each bracket equal to 0.

Worked example: 1) x squared + 3x + 4 = 2; 2) x squared + 3x + 2 = 0 (shift everything to one side of the equation); 3) (x+2)(x+1) = 0 (factorise); 4) x = −1, x = −2 (solve for when each bracket is equal to 0) — Worked example: x² + 3x + 4 = 2 → x² + 3x + 2 = 0 → (x+2)(x+1) = 0 → x = −1, x = −2.

Sometimes it can be difficult to factorise a quadratic in order to solve it. In this case, we use the trusty quadratic formula.

Now don't start to panic, the quadratic formula may seem daunting, but it will become your best friend in years to come!

The solutions to any quadratic will be given by substituting in the coefficients a, b, and the constant c into this formula.

The quadratic formula: x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a — The quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.

Still Struggling with Algebra?

Sometimes the internet just isn't enough and you really do need a mentor to guide you in person.

Smart Space offers in person lessons with engaging tutors. Maybe we can help you. Take a look at our Maths courses or book a free trial below.