Equating Coefficients

An identity is an algebraic statement with an identity sign, ≡, between a left-hand side and a right-hand side that is true for all values of x
- E.g. x + x ≡ 2x
- This means x + x is identical to 2x, or that x + x "can also be written as" 2x
An identity cannot be solved
All numbers can be substituted into an identity and it will remain true
- E.g. x + x ≡ 2x is true for x = 1, x = 2, x = 3 … (even x = -0.01, x = π etc)
- Unlike with equations, where only the solutions work
- E.g. 2x = 10 is not true for x = 1, x = 2, x = 3 … only for x = 5

Identities can be used to write algebraic expressions in different forms
For example, find p and q if 3(x + y) + 2y ≡ px + qy
- 3(x + y) + 2y expands to 3x + 5y
- The coefficient of x on the left is 3 and on the right is p, so p = 3
- The coefficient of y on the left is 5 and on the right is q, so q = 5
- Therefore 3(x + y) + 2y is identical to 3x + 5y
- This method is called equating coefficients

Given that $a {(x + 3)}^{2} - b x + 5 \equiv 4 x^{2} + c$ , find $a$ , $b$ and $c$ .

Expand the left hand side of the identity

$a (x + 3) (x + 3) - b x + 5 a (x^{2} + 6 x + 9) - b x + 5 a x^{2} + 6 a x + 9 a - b x + 5$

Group "like" terms

$a x^{2} + 6 a x - b x + 9 a + 5$

Compare "like" terms with the right hand side

$4 x^{2} + 0 x + c$

Equate the coefficients of the "like" terms to form three equations

$x^{2} : x : 1 :$ $a = 4 6 a - b = 0 9 a + 5 = c$

The first equation gives the value of a
Use this to find the value of b and c

$24 - b = 0 \Rightarrow b = 24 36 + 5 = c \Rightarrow c = 41$

$a = 4, b = 24, c = 41$

Equating Coefficients (AQA GCSE Further Maths)