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2a · E2.1–E2.4

Algebraic manipulation and indices

Topic 2 Algebra and graphs · Core and Extended · Papers 1–4

🎯What you need to be able to do

  • Substitute numbers into expressions and formulas; simplify by collecting like terms.
  • Expand brackets — two brackets for Core, three or more for Extended EXTENDED.
  • Factorise by taking out a common factor.
  • Factorise by grouping, the difference of two squares, perfect squares and quadratics \( ax^{2} + bx + c \) EXTENDED.
  • Complete the square EXTENDED.
  • Add, subtract, multiply, divide and simplify algebraic fractions EXTENDED.
  • Use the laws of indices with integer powers, and fractional powers for Extended EXTENDED; solve equations such as \( 2^{x} = 32 \).

📚The mathematics

Expanding brackets

Every term in one bracket multiplies every term in the other. A grid keeps the bookkeeping honest, and shows why \( (a + b)^{2} \ne a^{2} + b^{2} \): the two middle rectangles are the \( 2ab \) that people forget.

A rectangle split into four parts to expand (2x + 3)(x + 4): 2x times x is 2x squared, 2x times 4 is 8x, 3 times x is 3x, and 3 times 4 is 12, giving 2x squared plus 11x plus 12.
\( (2x + 3)(x + 4) = 2x^{2} + 8x + 3x + 12 = 2x^{2} + 11x + 12 \).

For three brackets EXTENDED, expand two first, simplify, then multiply by the third.

Factorising

“Factorise” means fully: always take out the highest common factor first.

  • Common factor: \( 9x^{2}y + 15xy^{2} = 3xy(3x + 5y) \).
  • EXTENDED Grouping: \( ax + bx + 2ay + 2by = x(a + b) + 2y(a + b) = (a + b)(x + 2y) \).
  • EXTENDED Difference of two squares: \( a^{2} - b^{2} = (a - b)(a + b) \), so \( 25x^{2} - 49y^{2} = (5x - 7y)(5x + 7y) \).
  • EXTENDED Quadratics \( ax^{2} + bx + c \): find two numbers that multiply to \( ac \) and add to \(b\), split the middle term, then group. For \( 6x^{2} - x - 12 \): \( ac = -72 \), and \( 8 \times (-9) = -72 \), \( 8 + (-9) = -1 \), so \( 6x^{2} + 8x - 9x - 12 = 2x(3x + 4) - 3(3x + 4) = (2x - 3)(3x + 4) \).
  • EXTENDED Cubics such as \( x^{3} - 5x^{2} + 6x \): take out \(x\), then factorise the quadratic: \( x(x - 2)(x - 3) \).
A large square of side a with a small square of side b removed from one corner. The L-shaped remainder is cut into two rectangles and rearranged into one rectangle measuring (a + b) by (a minus b), showing that a squared minus b squared equals (a + b)(a minus b).
Why \( a^{2} - b^{2} = (a + b)(a - b) \): the L-shape rearranges into a single rectangle.

Completing the square EXTENDED

Halve the coefficient of \(x\): \( x^{2} + bx + c = \left(x + \tfrac{b}{2}\right)^{2} - \left(\tfrac{b}{2}\right)^{2} + c \). So \( x^{2} - 6x + 11 = (x - 3)^{2} - 9 + 11 = (x - 3)^{2} + 2 \). The turning point of \( y = x^{2} - 6x + 11 \) is then \( (3, 2) \) — used again in 2d.

Algebraic fractions EXTENDED

Treat them exactly like number fractions. To add or subtract, use a common denominator and put every numerator in brackets (the minus sign applies to the whole of the second numerator). To simplify, factorise first, then cancel whole factors — never cancel single terms.

Indices

The rules are the ones from 1a, now with letters: \( (4x^{5})^{2} = 16x^{10} \), \( 18b^{6} \div 6b^{-3} = 3b^{9} \). To solve an equation such as \( 4^{x + 1} = 8^{x} \) EXTENDED, write both sides as powers of the same base, \( 2^{2x + 2} = 2^{3x} \), then equate the powers: \( x = 2 \). Logarithms are not required.

✏️Worked example (non-calculator, Extended)

(a) Expand and simplify \( (x - 2)(x + 3)(2x + 1) \). [3] (b) Simplify \( \dfrac{x^{2} - 9}{x^{2} + x - 6} \). [3] (c) Simplify \( \dfrac{(2x^{3}y)^{4}}{8x^{5}y^{2}} \). [2]

(a) \( (x - 2)(x + 3) = x^{2} + x - 6 \). Then \( (x^{2} + x - 6)(2x + 1) = 2x^{3} + x^{2} + 2x^{2} + x - 12x - 6 \), which is \( 2x^{3} + 3x^{2} - 11x - 6 \).

(b) \( \dfrac{(x - 3)(x + 3)}{(x + 3)(x - 2)} = \dfrac{x - 3}{x - 2} \).

(c) \( \dfrac{16x^{12}y^{4}}{8x^{5}y^{2}} = 2x^{7}y^{2} \).

Check it. Put \( x = 1 \) into (a): \( (-1)(4)(3) = -12 \) and \( 2 + 3 - 11 - 6 = -12 \) ✓. Put \( x = 4 \) into (b): \( \tfrac{7}{14} = \tfrac{1}{2} \) and \( \tfrac{1}{2} \) ✓.
Cancelling terms. In (b), crossing out the \( x^{2} \) on top and bottom gives \( \tfrac{-9}{x - 6} \), which is wrong. You may only cancel a factor of the whole numerator and the whole denominator.

📝Practise

Questions in the style of the current papers. EXTENDED marks Extended-only content.

1. (Non-calculator.) Expand and simplify \( (2x + 1)(x - 4) \). [2]
\( 2x^{2} - 8x + x - 4 = 2x^{2} - 7x - 4 \).
2. (Non-calculator.) Find the value of \( 3a^{2} - 2b \) when \( a = -4 \) and \( b = 5 \). [2]
\( 3(16) - 10 = 38 \). Square the \(-4\) first: \( (-4)^{2} = 16 \), not \(-16\).
3. (Non-calculator.) Simplify (a) \( (4x^{5})^{2} \), (b) \( 18b^{6} \div 6b^{-3} \). [2]
(a) \( 16x^{10} \). (b) \( 3b^{9} \).
4. (Non-calculator.) EXTENDED Factorise completely (a) \( 25x^{2} - 49y^{2} \), (b) \( ax + bx + 2ay + 2by \), (c) \( 3x^{2} + 10x - 8 \). [5]
(a) \( (5x - 7y)(5x + 7y) \). (b) \( (a + b)(x + 2y) \). (c) \( ac = -24 \); \( 12 \) and \( -2 \): \( 3x^{2} + 12x - 2x - 8 = 3x(x + 4) - 2(x + 4) = (3x - 2)(x + 4) \).
5. (Non-calculator.) EXTENDED Write \( x^{2} - 6x + 11 \) in the form \( (x - a)^{2} + b \). [2]
\( (x - 3)^{2} - 9 + 11 = (x - 3)^{2} + 2 \), so \( a = 3 \), \( b = 2 \).
6. (Non-calculator.) EXTENDED Write \( \dfrac{3}{x + 2} - \dfrac{1}{x - 1} \) as a single fraction in its simplest form. [3]
\( \dfrac{3(x - 1) - (x + 2)}{(x + 2)(x - 1)} = \dfrac{2x - 5}{(x + 2)(x - 1)} \). The bracket round \( (x + 2) \) is what makes the constant \(-2\), not \(+2\).
7. (Non-calculator.) EXTENDED Simplify \( \dfrac{2x^{2} - 8}{x^{2} + 5x + 6} \). [3]
\( \dfrac{2(x - 2)(x + 2)}{(x + 2)(x + 3)} = \dfrac{2(x - 2)}{x + 3} \).
8. (Non-calculator.) EXTENDED (a) Simplify \( (27x^{6})^{\frac{2}{3}} \). (b) Solve \( 25^{x} = 5^{x + 3} \). [4]
(a) \( 27^{\frac{2}{3}} = 9 \) and \( (x^{6})^{\frac{2}{3}} = x^{4} \): \( 9x^{4} \). (b) \( 5^{2x} = 5^{x + 3} \Rightarrow 2x = x + 3 \Rightarrow x = 3 \).

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • Corbettmaths — factorising quadratics where \( a \ne 1 \), and algebraic fractions
  • Desmos — graph an expression and its factorised form together to check they are identical