Welcome to the World of Algebra!

Welcome! Algebra is often described as the "language" of mathematics. Instead of just using fixed numbers, we use letters to represent values that can change. This allows us to solve complex problems, from predicting the path of a rocket to calculating interest on a savings account. Don't worry if it seems like a lot to take in—we'll break it down step-by-step, starting from the basics and moving into the Higher Tier skills you need for your Edexcel GCSE.

1. Algebraic Manipulation: The Building Blocks

Before we can solve big problems, we need to know how to "tidy up" our algebra. This is called simplifying.

Notation Rules

  • \(ab\) means \(a \times b\). We save time by leaving out the multiplication sign!
  • \(a^2\) means \(a \times a\).
  • \(\frac{a}{b}\) means \(a \div b\). Fractions are just division in disguise.
  • A coefficient is the number in front of a letter (e.g., in \(5x\), 5 is the coefficient). In the Higher Tier, coefficients are often fractions.

Expanding and Factorising

Think of expanding as "opening the brackets" and factorising as "putting them back in."

Expanding Triple Binomials

For the Higher Tier, you might need to expand more than two brackets, like \((x+1)(x+2)(x+3)\).
Step 1: Multiply the first two brackets together as usual.
Step 2: Take that result and multiply every term by the parts of the third bracket.

Factorising \(ax^2 + bx + c\)

When the number in front of \(x^2\) is greater than 1 (e.g., \(2x^2 + 7x + 3\)), it's a bit trickier.
Memory Aid: The "AC" Method. Multiply the first number (a) by the last number (c). Find two numbers that multiply to give that result but add to give the middle number (b).

Algebraic Fractions

Treat these just like normal fractions! To add or subtract them, you need a common denominator. To simplify them, factorise the top and bottom first and see if anything "cancels out."

Quick Review: Always look for a Highest Common Factor (HCF) first when factorising. It's the easiest way to start!

2. Functions: The Mathematical Machines

A function is like a machine: you put a number in (the input), it does something to it, and a number comes out (the output).

Composite Functions

This is when you use two machines in a row. Notation: \(fg(x)\).
Important: You work from right to left. Calculate \(g(x)\) first, then put that answer into \(f\).

Inverse Functions

The inverse function, written as \(f^{-1}(x)\), "undoes" the original function. It’s like the "rewind" button.
Step-by-step: To find an inverse, write the function as \(y = ...\), swap the \(x\) and \(y\), then rearrange it to get \(y\) on its own again.

Key Takeaway: \(fg(x)\) means "do \(g\) first, then \(f\)." \(f^{-1}(x)\) means "reverse the process."

3. Solving Equations and Inequalities

Solving means finding the specific value of the letter that makes the statement true.

Quadratic Equations

For equations like \(ax^2 + bx + c = 0\), you have three main weapons:

  1. Factorising: Great if the numbers are simple.
  2. The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This works for any quadratic!
  3. Completing the Square: Writing the equation in the form \((x+p)^2 + q\). This is very useful for finding the turning point of a graph.

Simultaneous Equations

At the Higher Tier, you will solve one linear and one quadratic equation together (e.g., \(y = x + 2\) and \(x^2 + y^2 = 10\)).
Top Tip: Always use substitution. Replace the \(y\) in the quadratic equation with the expression from the linear one.

Iteration

Sometimes equations are too hard to solve exactly. We use iteration to find an approximate answer by doing the same calculation over and over, getting closer to the answer each time. Usually, the question will give you a formula like \(x_{n+1} = \sqrt{x_n + 5}\).

Common Mistake: When solving inequalities like \(-3x < 9\), if you divide by a negative number, you must flip the inequality sign! (\(x > -3\)).

4. Graphs: Seeing the Algebra

Graphs let us visualize equations. You need to recognize several shapes:

  • Linear: \(y = mx + c\) (A straight line).
  • Quadratic: \(y = x^2\) (A "U" or "n" shaped curve called a parabola).
  • Cubic: \(y = x^3\) (An "S" shaped curve).
  • Reciprocal: \(y = \frac{1}{x}\) (Curves that never touch the axes).
  • Exponential: \(y = k^x\) (Starts slow, then shoots up rapidly).

Circle Equations

The equation of a circle with its centre at \((0,0)\) is \(x^2 + y^2 = r^2\), where \(r\) is the radius.
Did you know? If you see \(x^2 + y^2 = 25\), the radius is simply \(\sqrt{25} = 5\).

Gradients and Tangents

For a curve, the gradient changes at every point. A tangent is a straight line that just touches the curve at one point. The gradient of the tangent is the "instantaneous rate of change" at that exact spot.

Transforming Graphs

If we have a graph \(y = f(x)\):

  • \(f(x) + k\): Shifts the graph up.
  • \(f(x - k)\): Shifts the graph right (remember, inside the bracket is the opposite of what you'd expect!).
  • \(-f(x)\): Reflects the graph in the x-axis (upside down).

Key Takeaway: Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to give -1.

5. Sequences

A sequence is just a list of numbers following a pattern.

Arithmetic and Geometric

  • Arithmetic: You add or subtract the same number every time (e.g., 2, 5, 8...).
  • Geometric: You multiply by the same number (the common ratio \(r\)) every time (e.g., 3, 6, 12...).

Quadratic Sequences

In these sequences, the difference between the numbers changes, but the second difference is constant.
The \(n\)th term formula looks like: \(an^2 + bn + c\).
Trick: The second difference is always equal to \(2a\). So, if the second difference is 4, then \(a = 2\)!

Fibonacci Sequences

In a Fibonacci-type sequence, you find the next term by adding the two previous terms (e.g., 1, 1, 2, 3, 5, 8...).

Quick Review: The \(n\)th term allows you to find any position in a sequence. To find the 100th term, just replace \(n\) with 100 in your formula!

Final Words of Encouragement

Algebra can feel like a puzzle. Sometimes you might get stuck, and that's perfectly okay! The key is to keep practicing the "moves"—expanding, factorising, and rearranging. Once you master those, the harder problems become much easier to solve. You've got this!