AQA · Mathematics · 8300
GCSE Maths: Identities: Equating coefficients
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths practice questions for Identities: Equating coefficients (Corbettmaths Video 367), and every card links back to the exact page it came from. Practice tests generate fresh variations of these questions, so the numbers change every attempt. Source material © Corbettmaths — used with attribution.
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What does the symbol ≡ mean between two algebraic expressions?
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It is an identity: the two expressions are equal for every value of the variable, so matching coefficients must be equal.
How do you solve for unknowns in an identity like 5(3x + c) ≡ 15x + 40?
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Expand the left side, then equate the coefficients of like terms (x-terms with x-terms, constants with constants) and solve.
If 5(3x + c) ≡ 15x + 40, what is c?
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c = 8, because 5c = 40.
If 2(x + 7) + cx + d ≡ 5x + 1, what are c and d?
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c = 3 and d = −13 (x: 2 + c = 5; constants: 14 + d = 1).
If a(2x + 7) ≡ 8x + 14b, what are a and b?
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a = 4 and b = 2 (2a = 8; 7a = 28 = 14b).
If 9(2x + 5) + 2(3x + b) ≡ ax + 23, what are a and b?
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a = 24 and b = −11 (LHS = 24x + 45 + 2b, so 45 + 2b = 23).
If 4(3x + 2) + a(x + b) ≡ 15x − 16, what are a and b?
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a = 3 and b = −8 (x: 12 + a = 15; constants: 8 + ab = −16 → 3b = −24).
If (x + 4)(x − 6) + ax + b ≡ x² + 8x − 25, what are a and b?
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a = 10 and b = −1 (expansion gives x² − 2x − 24, so a − 2 = 8 and b − 24 = −25).
If (x + b)² ≡ x² + cx + 49, what are the two possible values of c?
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c = 14 or c = −14, because b² = 49 gives b = ±7 and c = 2b.
If (x + 2c)(x + c) ≡ x² + dx + 50, what are the two possible values of d?
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d = 15 or d = −15, because 2c² = 50 gives c = ±5 and d = 3c.
If (ax + 1)(x + 5)(x + b) ≡ 2x³ + 23x² + 71x + 30, what are a and b?
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a = 2 and b = 6 (x³ coefficient: a = 2; constant term: 1 × 5 × b = 30).
Which two coefficients are quickest to compare in a cubic identity such as (ax + 1)(x + 5)(x + b) ≡ 2x³ + 23x² + 71x + 30?
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The x³ coefficient (product of the leading terms) and the constant term (product of the constants) — neither needs a full expansion.
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