📐 Algebra & Equations
Algebra is the highest-weightage topic in E-Math, appearing in both papers and forming the foundation for many other topics including graphs, trigonometry, and statistics. You must master algebraic manipulation, factorisation, quadratic equations, simultaneous equations, algebraic fractions, and inequalities. This chapter provides comprehensive coverage with multiple worked examples for every key technique.
Algebraic fluency is what separates A1 students from the rest. Many students understand the concepts but lose marks because they make errors during manipulation. The key to mastery is practice — work through every example in this chapter, then attempt past paper questions until the techniques become second nature.
Algebraic Expansion & Identities
Expansion means removing brackets by multiplying each term inside the bracket by the term outside. For two binomials, use the FOIL method (First, Outer, Inner, Last) or the grid method. However, the three special algebraic identities below are so commonly used that you should memorise them and apply them directly — this saves time and reduces errors.
Perfect Square (Sum)
Perfect Square (Difference)
Difference of Two Squares
The most common mistake with perfect squares is forgetting the middle term. Students write , which is WRONG. The correct expansion always has the middle term 2ab. Always check your expansion by counting terms: a perfect square expansion should have 3 terms.
🧮Algebraic Expansion & Factorisation
Special algebraic identities tested frequently in Paper 1. Must be able to expand and factorise fluently.
📝Worked Example 1: Using Algebraic Identities
Expand and simplify .
📝Worked Example 2: Expanding Three Brackets
Expand .
Expand
Factorisation
Factorisation is the reverse of expansion — you start with an expanded expression and rewrite it as a product of factors. This is a crucial skill because it is required for solving quadratic equations, simplifying algebraic fractions, and sketching graphs. There are four main methods of factorisation, and you should try them in order until one works.
4 Methods of Factorisation (Try in This Order)
1. Common factor
Always check for a common factor first. Example: . Look for the LARGEST common factor of all terms.
2. Grouping
For expressions with 4 terms. Group in pairs and factor each pair. Example: .
3. Difference of two squares
For expressions of the form . Example: . Only works for SUBTRACTION of perfect squares.
4. Quadratic trinomial (3 terms)
For . Find two numbers that multiply to and add to . Split the middle term and factor by grouping.
📝Worked Example 3: Common Factor
Factorise completely .
📝Worked Example 4: Factorising a Quadratic Trinomial (a = 1)
Factorise .
📝Worked Example 5: Factorising a Quadratic Trinomial (a > 1)
Factorise .
📝Worked Example 6: Difference of Two Squares
Factorise and hence evaluate without a calculator.
Factorise:
Algebraic Fractions
Algebraic fractions follow the same rules as numerical fractions. The most common operation tested is addition and subtraction, which requires finding a common denominator. Always factorise the denominators first to find the LCD (Lowest Common Denominator). When multiplying, cancel common factors before multiplying. When dividing, flip the second fraction and multiply.
📝Algebraic Fractions
Simplify, add, subtract, multiply, and divide algebraic fractions.
📝Worked Example 7: Adding Algebraic Fractions
Simplify .
📝Worked Example 8: Subtracting Algebraic Fractions
Simplify .
📝Worked Example 8b: Multiplying Algebraic Fractions
Simplify .
📝Worked Example 8c: Solving an Equation with Algebraic Fractions
Solve .
📝Worked Example 8d: Algebraic Fraction Leading to Quadratic
Solve .
Simplify:
Quadratic Equations — 3 Methods
A quadratic equation has the form where . There are three methods to solve it, and you should know all three because the exam may specify which method to use, or certain equations are better suited to particular methods.
📐Quadratic Equations
Three methods to solve quadratic equations: factorisation, completing the square, and the quadratic formula.
Method 1: Factorisation
When to use: When the quadratic factorises neatly with integer roots. This is the fastest method — try it first. Look for two numbers that multiply to give and add to give .
📝Worked Example 9: Factorisation
Solve by factorisation.
Method 2: Quadratic Formula
When to use: When the equation does not factorise neatly, or when the question asks for answers to a given number of significant figures or decimal places. This method always works for any quadratic.
The Quadratic Formula
📝Worked Example 10: Rational Roots
Solve using the quadratic formula.
📝Worked Example 11: Irrational Roots (3 s.f.)
Solve , giving answers correct to 3 significant figures.
Method 3: Completing the Square
When to use: When the question explicitly says “express in the form ”, or when you need to find the turning point of a quadratic graph. Also useful for proving results.
📝Worked Example 12: Completing the Square
Express in the form , then solve .
Discriminant Quick Guide
The discriminant tells you the nature of the roots before you solve:
- D > 0: Two distinct real roots (if D is a perfect square, roots are rational)
- D = 0: Two equal real roots (the parabola just touches the x-axis)
- D < 0: No real roots (the parabola does not cross the x-axis)
Solve:
Simultaneous Equations
Simultaneous equations involve finding the values of two (or more) unknowns that satisfy multiple equations at the same time. For two linear equations, you can use either the elimination method or the substitution method. If one equation is linear and one is non-linear (e.g., one is quadratic), you must use substitution.
🔗Simultaneous Equations
Solve a pair of linear equations or one linear + one non-linear equation.
📝Worked Example 13: Elimination Method
Solve: and .
📝Worked Example 14: Elimination with Multiplication
Solve: and .
📝Worked Example 15: Substitution Method
Solve: and .
📝Worked Example 16: Simultaneous with Quadratic
Solve: and .
Inequalities
Linear inequalities are solved using the same techniques as linear equations, with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. This is the number one error students make with inequalities. Always be extra careful when negative coefficients are involved.
📏Inequalities
Solve linear inequalities and represent solutions on a number line.
Critical Rule: Flipping the Inequality Sign
When you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality sign. For example: if $-3x > 6$, dividing by $-3$ gives $x < -2$ (sign flipped). This is the #1 error students make.
📝Worked Example 17: Solving an Inequality
Solve and represent the solution on a number line.
📝Worked Example 18: Compound Inequality
Find the integer values of x satisfying .
Solve:
⚠️Common Mistakes — Algebra
Why: Must use the identity (a+b)^2 = a^2 + 2ab + b^2. The middle term 2ab = 2(x)(3) = 6x cannot be omitted.
Why: When distributing a negative sign, ALL terms inside change sign. Negative times negative gives positive.
Why: The difference of two squares works for subtraction only. Sum of two squares cannot be factorised over the reals.
Why: The entire expression is divided by 2a, not just 2. The denominator is 2a. Missing the "a" in the denominator is a very common error.
Why: Always write plus/minus before the square root. Forgetting gives only one root instead of two.
Why: When you multiply or divide both sides of an inequality by a negative number, the inequality sign MUST be reversed.
Why: Algebraic fractions require a common denominator before adding. You cannot simply add numerators and denominators.
Changing the Subject of a Formula
"Changing the subject" means rearranging a formula to isolate a different variable. This is tested frequently and requires confident algebraic manipulation. The key is to perform the same operation on both sides, working outward from the variable you want to isolate.
📝Worked Example 19: Changing the Subject (Linear)
Make r the subject of .
📝Worked Example 20: Changing Subject with Fraction
Make t the subject of .
📝Worked Example 21: Subject with Variable on Both Sides
Make x the subject of .
📝Worked Example 21b: Subject with Square Root
Make g the subject of .
📝Worked Example 21c: Subject Appears in Two Places
Make y the subject of .
Make the subject:
Forming Equations from Geometric Contexts
A very common exam question type gives you a geometric diagram with dimensions expressed in terms of x, and asks you to form an equation and solve it. These questions combine algebra with geometry knowledge (perimeter, area, angles, Pythagoras). The key skill is translating the geometric relationship into an algebraic equation.
📝Worked Example 21d: Rectangle Area Problem
A rectangle has length cm and width cm. Its area is 65 cm squared. Find the value of x and hence the dimensions of the rectangle.
📝Worked Example 21e: Forming Equation from Triangle Angles
A triangle has angles (3x + 10) degrees, (2x + 30) degrees, and (x + 20) degrees. Find x and hence state all three angles.
📝Worked Example 21f: Perimeter and Pythagoras
A right-angled triangle has legs of length cm and cm, and hypotenuse cm. Find x.
The angles of a quadrilateral are x, 2x, 3x, and 4x degrees. Find x.
Algebraic Word Problems
Word problems require you to translate a real-world situation into algebraic equations, then solve them. The steps are: (1) Define your variables clearly, (2) Write equations from the given information, (3) Solve the equations, (4) Interpret and check your answer in context.
📝Worked Example 22: Age Problem
Ahmad is 3 times as old as his son. In 12 years, Ahmad will be twice as old as his son. Find their current ages.
📝Worked Example 23: Number Problem Leading to Quadratic
The product of two consecutive positive integers is 182. Find the integers.
📝Worked Example 24: Speed Problem Leading to Equation
A car travels 240 km. If it had been 20 km/h faster, the journey would have taken 1 hour less. Find the speed of the car.
📝Worked Example 25: Work Rate Problem
Ali can paint a room in 6 hours. Beng can paint the same room in 4 hours. How long will it take if they work together?
📝Worked Example 26: Profit and Loss Word Problem
A shopkeeper buys an item for $x. She marks it up by 60% and then offers a 25% discount. She makes a profit of $14. Find x.
Ali completes a job in 5 hours. Bala completes it in 10 hours. How long working together?
Algebra Study Plan
Spend at least 40% of your E-Math study time on algebra. Start with expansion and factorisation (the foundation), then move to quadratic equations, simultaneous equations, and algebraic fractions. Practise at least 10 past paper algebra questions before the exam. If you can handle algebra confidently, many other topics (graphs, coordinate geometry) become much easier.