Algebraic Proof (AQA GCSE Further Maths) : Revision Note

Algebraic Proof

What is algebraic proof?

• Algebraic Proof is the process of showing something is true in every case, using algebra

• Typical algebra skills include expanding brackets and collecting like terms

  • At the harder end, knowing the "difference of two squares" factorisation is useful

How do I prove results about odd and even numbers?

• Assign letters (use as few letters as possible):

• n is “any integer” (or m or k or…)

  • Integer means whole number

• n + 1 is the consecutive integer after n (the one immediately after n)

• 2n is an even integer (2n + 2 is the next one)

• 2m is a different even integer (not necessarily consecutive, but any other even integer)

• 2n + 1 is an odd integer (and 2n + 3 is the next one, or 2n - 1 is the one before, etc)

• A "multiple of k” means it can be written as k(……), ie. k × …

• To prove something is even, show that the algebraic result can be written as 2 × (...)

  • Make sure whatever is inside the brackets is an integer

• To prove something is odd, show that the algebraic result can be written as 2 × (...) + 1

  • Make sure whatever is inside the brackets is an integer

• When dealing with prime numbers, remember that primes only have factors of 1 and themselves

  • If p is prime then 1 × p or p × 1 are the only ways to write it as a product of two integers

What is the difference between an equation and an identity?

• An equation is true for certain values only

  • For example, 3x − 1 = 5 is an equation and is only true when x = 2

  • Or another example, x² = 9 is an equation and is true only when x = 3 or when x = −3

• An identity is true for all values

  • For example, 2(3x) ≡ 6x is an identity because it is true for all values of x

    • Note that the symbol for an identity, ≡, is 3 horizontal lines (like an equals sign but with an extra line)

How can I use completing the square to prove something is positive?

• Squaring anything makes it positive...

  • ...unless its zero (which squares to zero)

• Substituting any value of x into the expression (x - 2)² will always give a positive output due to the "squared" outside the brackets...

  • unless the bit inside the brackets equals zero

    • x - 2 = 0, i.e. x = 2 

    • in which case, substituting in x = 2 gives zero

• Completing the square helps to show an expression is always positive

  • f(x) = x² - 6x + 11 can be written f(x) = (x - 3)² + 2

    • (x - 3)²  ≥ 0 

    • (x - 3)² + 2 ≥  2

    • so all outputs are positive (in fact, they're greater than or equal to 2)