Algebra & Identities • Topic 1 of 4

Algebraic Identities

The square, cube and difference-of-squares identities turn long multiplications into one line. Recognise the shape: a^2 + b^2 = (a+b)^2 - 2ab; a^3 + b^3 = (a+b)^3 - 3ab(a+b). The symmetric identity a^3 + b^3 + c^3 = 3abc whenever a + b + c = 0 is a frequent SSC shortcut. Always look for what is given (a sum, a product) and choose the identity that uses exactly those.

✅ Solved examples

1. If a + b = 7 and ab = 12, find a^2 + b^2.
(a+b)^2 - 2ab = 49 - 24 = 25.
2. Evaluate 105^2 - 95^2.
(105+95)(105-95) = 200 x 10 = 2000.
3. If a + b = 5 and ab = 6, find a^3 + b^3.
(a+b)^3 - 3ab(a+b) = 125 - 3x6x5 = 125 - 90 = 35.
4. If a + b + c = 0, find a^3 + b^3 + c^3 given abc = 4.
When sum = 0, a^3+b^3+c^3 = 3abc = 12.

✏️ Practice — try these, take hints as needed

1. a + b = 9, ab = 20. Find a^2 + b^2.
(a+b)^2 - 2ab.
81 - 40.
—
41
2. Evaluate 53^2 - 47^2.
(53+47)(53-47).
100 x 6.
—
600
3. a - b = 3, ab = 10. Find a^2 + b^2.
(a-b)^2 + 2ab.
9 + 20.
—
29
4. a + b = 4, ab = 3. Find a^3 + b^3.
(a+b)^3 - 3ab(a+b).
64 - 36.
—
28
5. If x + y + z = 0 and xyz = 5, find x^3+y^3+z^3.
= 3xyz.
3 x 5.
—
15

📝 Topic test — 8 questions

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