Equations, identities and formulae
Edexcel 1MA1 explicitly tests the difference between an equation, an identity and a formula on Higher tier (A6 in the spec). Foundation may test "show that" using simple substitution.
📖Definition— Definitions
- Equation — equality that holds for specific values of the variable. Solve to find them. e.g. 2x + 3 = 11 has solution x = 4.
- Identity — equality that holds for all values of the variable. Use the symbol ≡. e.g. (x + 2)² ≡ x² + 4x + 4.
- Formula — a general relationship between two or more variables. e.g. A = πr² gives the area for any radius.
- Expression — collection of terms with no equals sign. e.g. 3x + 5.
Edexcel notation
The triple-bar ≡ is used in identities. Mark schemes accept = if the question doesn't explicitly require ≡, but using ≡ when working with identities scores the C1 mark.
"Show that" identities
A typical Higher A6 question: "Show that (x + 3)(x + 4) − x(x + 1) ≡ 6x + 12."
Method:
- Expand both products. (x + 3)(x + 4) = x² + 7x + 12. x(x + 1) = x² + x.
- Subtract: x² + 7x + 12 − x² − x = 6x + 12.
- Conclude with ≡ on the final line.
Mark scheme: M1 expand both, M1 subtract correctly, A1 reach 6x + 12 with ≡ stated.
Finding unknown coefficients
A common Edexcel format: "Find values of a and b so that a(x + 1)² + b ≡ 2x² + 4x + 9."
Method:
- Expand: a(x² + 2x + 1) + b = ax² + 2ax + a + b.
- Compare coefficients: a = 2, 2a = 4 (consistent), a + b = 9 → b = 7.
Mark scheme: M1 expand, M1 equate coefficients, A1 a = 2, A1 b = 7.
Algebraic equivalence — Higher proof style
To prove two expressions are equivalent, simplify both to the same form, or simplify the difference to zero. Always finish with the symbol ≡ on the last line.
Common Edexcel exam tip
The most common mistake is using "=" where "≡" is required. The mark scheme often awards a C1 specifically for the correct ≡ symbol on the conclusion line. It costs nothing — write it.
⚠Common mistakes— Common errors
- Mixing up identity and equation: "x² + 4x + 4" is an expression, "(x+2)² ≡ x² + 4x + 4" is an identity, "x² + 4x + 4 = 0" is an equation (with double-root x = −2).
- Forgetting that an identity holds for all x; substituting a single value to "verify" is not a proof.
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