Introduction to Equations

Welcome to the world of equations! In Algebra, an equation is a mathematical statement that shows two expressions are equal using an equals sign \( = \). You can think of an equation like a balance scale: whatever you do to one side, you must do to the other to keep it level.

In this chapter, we will explore linear (1st degree), quadratic (2nd degree), and cubic (3rd degree) equations. Mastering these is like learning the "rules of the game" for the rest of your International GCSE Mathematics course.

1. Linear Equations (1st Degree)

A linear equation is the simplest type of equation. The highest power of the unknown variable (usually \(x\)) is \(1\). They look like this: \(ax + b = c\).

Goal: Isolate the variable (get \(x\) by itself).

Step-by-Step Solving:

  1. Expand any brackets if they exist.
  2. Collect all terms containing \(x\) on one side and numbers on the other.
  3. Simplify both sides.
  4. Divide by the coefficient (the number in front) of \(x\).

Example: Solve \(4(x - 3) = 2x + 10\)
1. Expand: \(4x - 12 = 2x + 10\)
2. Subtract \(2x\) from both sides: \(2x - 12 = 10\)
3. Add \(12\) to both sides: \(2x = 22\)
4. Divide by \(2\): \(x = 11\)

Key Takeaway: Always perform the inverse operation (the opposite). If a number is added, subtract it; if it is multiplied, divide it!

2. Quadratic Equations (2nd Degree)

A quadratic equation contains a squared term, such as \(x^2\). The standard form is \(ax^2 + bx + c = 0\). Because of the \(x^2\), these equations usually have two solutions (roots).

The syllabus requires you to know four ways to solve these:

Method A: Factorisation

If you can write the equation as \((x + p)(x + q) = 0\), then either \(x + p = 0\) or \(x + q = 0\). This means \(x = -p\) or \(x = -q\).

Note: For a refresher on how to factorise, see the chapter on "Factorisation".

Method B: Using the Quadratic Formula

This is a "fail-safe" method that works for any quadratic. The formula is provided on your Formula Sheet, so you don't need to memorise it, but you must know how to use it!

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Common Mistake: Be very careful with negative signs, especially if \(b\) or \(c\) are negative numbers!

Method C: Completing the Square

This involves rewriting \(x^2 + bx + c = 0\) in the form \((x + p)^2 + q = 0\).
1. Take half of the coefficient of \(x\).
2. Write it as \((x + \frac{b}{2})^2\).
3. Subtract the square of that same number: \((x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c = 0\).

Method D: Using Graphs

You can solve \(ax^2 + bx + c = 0\) by drawing the graph of \(y = ax^2 + bx + c\) and finding where the curve crosses the x-axis (where \(y = 0\)).

Did you know? The shape of a quadratic graph is called a parabola. It looks like a "U" or an upside-down "n".

3. Cubic Equations (3rd Degree)

Cubic equations have a highest power of \(x^3\). They take the form \(ax^3 + bx^2 + cx + d = 0\). These can have up to three solutions.

Solving by Factor Theorem

At this level, you usually solve cubics by finding one "easy" root first using the Factor Theorem.

  1. Try small integer values for \(x\) (like \(1, -1, 2, -2\)) to see which one makes the equation equal zero.
  2. If \(x = k\) makes the equation \(0\), then \((x - k)\) is a factor.
  3. Use Algebraic Division to divide the cubic by \((x - k)\). This leaves you with a quadratic equation.
  4. Solve the remaining quadratic using the methods mentioned in Section 2.

Note: See the "Factorisation, factor theorem and algebraic division" chapter for detailed steps on division.

Solving by Graphs

You may be asked to draw a cubic graph. The solutions to the equation are the points where the curve intersects the x-axis. If you have a curve \(y = f(x)\) and a line \(y = k\), the solutions to \(f(x) = k\) are the x-coordinates of the intersection points.

4. Problems Leading to Equations

The exam will often give you a word problem or a geometric shape and ask you to find a missing value. This requires constructing an equation first.

Step-by-Step Approach:

  1. Identify what the question is asking for and label it \(x\).
  2. Use the information given (e.g., "the area of the rectangle is \(50cm^2\)") to write an expression.
  3. Set the expression equal to the value given to form your equation.
  4. Solve the equation and check if your answer makes sense (e.g., a length cannot be negative!).

Quick Review Box:
- Linear: Power of \(1\). Use inverse operations.
- Quadratic: Power of \(2\). Use formula, factorising, or completing the square.
- Cubic: Power of \(3\). Use factor theorem or graphs.

Common Mistakes to Avoid

  • Forgetting the \(\pm\): When taking a square root, remember there is a positive and a negative possibility.
  • The "Zero" Rule: For quadratics and cubics, you must make the equation equal zero before you try to factorise or use the formula.
  • Sign Errors: Multiplying two negatives makes a positive. This is the most common place to lose marks in the quadratic formula!
  • Units: In "real-world" problems, don't forget to include units (like \(cm\) or \(kg\)) in your final answer if required.