Algebraic Manipulation Masterclass
1. The Basics: Terms & Brackets
Collecting Like Terms
You can only add or subtract terms that have exactly the same letters and powers.
Simplify: \( 2x^2 + 5x - x^2 + 3y + 2x \)
Group \(x^2\): \(2x^2 - x^2 = x^2\)
Group \(x\): \(5x + 2x = 7x\)
Group \(y\): \(3y\) (Alone)
Result: \( x^2 + 7x + 3y \)
Multiplying Single Terms
Multiply the term outside by everything inside.
Expand: \( 3x(2x - 5) \)
\( 3x \times 2x = 6x^2 \)
\( 3x \times -5 = -15x \)
Result: \( 6x^2 - 15x \)
Taking Out Common Factors
This is the reverse of expanding. Find the Highest Common Factor (HCF) of numbers and letters.
Factorise: \( 6x^2 + 9x \)
HCF of numbers (6, 9) is 3.
HCF of letters (\(x^2, x\)) is \(x\).
Pull \(3x\) outside.
Result: \( 3x(2x + 3) \)
2. Expanding Binomials
Double Brackets
Use the FOIL method (First, Outside, Inside, Last).
Expand: \( (x + 3)(x - 4) \)
First: \(x^2\)
Outside: \(-4x\)
Inside: \(+3x\)
Last: \(-12\)
Simplify: \(x^2 - 4x + 3x - 12\)
Result: \( x^2 - x - 12 \)
Triple Brackets
Expand two brackets first, simplify, then multiply by the third.
Expand: \( (x+1)(x+2)(x+3) \)
1. \((x+1)(x+2) = x^2 + 3x + 2\)
2. Multiply by \((x+3)\) → \( (x^2 + 3x + 2)(x+3) \)
Multiply everything by \(x\): \(x^3 + 3x^2 + 2x\)
Multiply everything by \(3\): \(+3x^2 + 9x + 6\)
Result: \( x^3 + 6x^2 + 11x + 6 \)
3. Factorising Quadratics
Type 1: \( x^2 + bx + c \)
Find two numbers that multiply to make \(c\) and add to make \(b\).
Factorise \( x^2 + 7x + 10 \)
Multiplies to 10: (1,10) or (2,5).
Adds to 7: 2 and 5 work.
Result: \( (x+2)(x+5) \)
Type 2: Difference of Two Squares (DOTS)
If you have \( (\text{Square}) - (\text{Square}) \), it factorises to \( (a+b)(a-b) \).
Factorise \( x^2 - 16 \)
\(\sqrt{x^2} = x\), \(\sqrt{16} = 4\).
Result: \( (x+4)(x-4) \)
Type 3: \( ax^2 + bx + c \) (Harder Quadratics)
The "AC Method": Multiply \(a \times c\). Find factors of that number that add to \(b\). Split the middle term.
Factorise \( 2x^2 + 7x + 3 \)
\( a \times c = 2 \times 3 = 6 \).
Factors of 6 that add to 7? (6 and 1).
Split \(7x\) into \(6x + 1x\):
\( 2x^2 + 6x + x + 3 \)
Factorise in pairs: \( 2x(x+3) + 1(x+3) \)
Result: \( (2x+1)(x+3) \)
4. Laws of Indices & Surds
Index Laws:
- Multiply: \( x^a \times x^b = x^{a+b} \)
- Divide: \( x^a \div x^b = x^{a-b} \)
- Brackets: \( (x^a)^b = x^{ab} \)
- Negative: \( x^{-a} = \frac{1}{x^a} \)
Simplify \( (2x^3)^4 \)
Apply power to the number: \(2^4 = 16\).
Apply power to the algebra: \( (x^3)^4 = x^{12} \).
Result: \( 16x^{12} \)
Surds
Treat surds like algebra. \(\sqrt{a} \times \sqrt{a} = a\).
Expand \( (2 + \sqrt{3})^2 \)
\((2 + \sqrt{3})(2 + \sqrt{3})\)
\( 4 + 2\sqrt{3} + 2\sqrt{3} + 3 \)
Result: \( 7 + 4\sqrt{3} \)
5. Algebraic Fractions
To simplify, you must factorise top and bottom first, then cancel common brackets.
Simplify \( \frac{x^2 - 9}{x^2 + 5x + 6} \)
Top (Difference of squares): \( (x+3)(x-3) \)
Bottom (Quadratic): \( (x+3)(x+2) \)
Cancel the \((x+3)\).
Result: \( \frac{x-3}{x+2} \)
Practice Questions & Solutions
Click on the question to verify your answer.
Q1: Expand and simplify \( 4(x + 5) + 2(3x - 1) \)
Expand brackets: \( 4x + 20 + 6x - 2 \)
Collect like terms: \( 4x + 6x = 10x \) and \( 20 - 2 = 18 \)
Answer: \( 10x + 18 \)
Q2: Factorise \( 8x^2 - 12x \)
HCF of 8 and 12 is 4.
HCF of \(x^2\) and \(x\) is \(x\).
Factor is \(4x\).
Answer: \( 4x(2x - 3) \)
Q3: Factorise \( x^2 - 2x - 15 \)
Multiply to -15, Add to -2.
Factors of -15: (-5, 3) works because \(-5 + 3 = -2\).
Answer: \( (x - 5)(x + 3) \)
Q4: Factorise fully \( 3x^2 - 27 \)
Step 1: Take out common factor 3.
\( 3(x^2 - 9) \)
Step 2: Spot the Difference of Two Squares inside.
Answer: \( 3(x + 3)(x - 3) \)
Q5: Factorise \( 3x^2 + 10x + 8 \)
\( a \times c = 3 \times 8 = 24 \).
Find factors of 24 that add to 10: (6 and 4).
Split middle term: \( 3x^2 + 6x + 4x + 8 \)
Factorise pairs: \( 3x(x+2) + 4(x+2) \)
Answer: \( (3x + 4)(x + 2) \)
Q6: Simplify \( \frac{2x^3 \times 6x^5}{4x^2} \)
Top: \( 2 \times 6 = 12 \) and \( x^3 \times x^5 = x^8 \).
Expression is \( \frac{12x^8}{4x^2} \).
Divide numbers: \( 12 \div 4 = 3 \).
Subtract indices: \( 8 - 2 = 6 \).
Answer: \( 3x^6 \)
Q7: Expand \( (2x + 1)(x - 3)(x + 2) \)
1. \((2x+1)(x-3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\)
2. Multiply by \((x+2)\).
\( x(2x^2 - 5x - 3) = 2x^3 - 5x^2 - 3x \)
\( 2(2x^2 - 5x - 3) = 4x^2 - 10x - 6 \)
3. Collect: \( 2x^3 - x^2 - 13x - 6 \)
Answer: \( 2x^3 - x^2 - 13x - 6 \)
Q8: Simplify \( \sqrt{75} + \sqrt{12} \)
Simplify \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).
Simplify \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\).
Add: \( 5\sqrt{3} + 2\sqrt{3} \).
Answer: \( 7\sqrt{3} \)
Q9: Simplify the algebraic fraction \( \frac{x^2 - x - 20}{x^2 - 16} \)
Factorise Top: Adds to -1, Multiplies to -20 → \((x-5)(x+4)\).
Factorise Bottom (DOTS): \((x+4)(x-4)\).
Cancel the \((x+4)\).
Answer: \( \frac{x-5}{x-4} \)
Q10: Express as a single fraction \( \frac{3}{x+1} + \frac{2}{x-3} \)
Common denominator is \((x+1)(x-3)\).
Cross multiply tops: \( 3(x-3) + 2(x+1) \)
Expand top: \( 3x - 9 + 2x + 2 = 5x - 7 \)
Answer: \( \frac{5x - 7}{(x+1)(x-3)} \)