Algebra & Identities • Topic 2 of 4

x + 1/x Problems

These symmetric problems are pure identity application. Given x + 1/x = k: x^2 + 1/x^2 = k^2 - 2, and x^3 + 1/x^3 = k^3 - 3k. For the minus family, x - 1/x = m gives x^2 + 1/x^2 = m^2 + 2. If you are given x^2 + 1/x^2, work back to x + 1/x by adding 2 and taking the root.

✅ Solved examples

1. If x + 1/x = 4, find x^2 + 1/x^2.

4^2 - 2 = 14.

2. If x + 1/x = 3, find x^3 + 1/x^3.

3^3 - 3x3 = 27 - 9 = 18.

3. If x - 1/x = 2, find x^2 + 1/x^2.

2^2 + 2 = 6.

4. If x^2 + 1/x^2 = 23, find x + 1/x.

(x + 1/x)^2 = 23 + 2 = 25 -> x + 1/x = 5.

✏️ Practice — try these, take hints as needed

1. x + 1/x = 5. Find x^2 + 1/x^2.

k^2 - 2.

25 - 2.

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2. x + 1/x = 2. Find x^3 + 1/x^3.

k^3 - 3k.

8 - 6.

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3. x - 1/x = 3. Find x^2 + 1/x^2.

m^2 + 2.

9 + 2.

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4. x^2 + 1/x^2 = 34. Find x + 1/x.

Add 2, take root.

root 36.

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5. x + 1/x = 6. Find x^3 + 1/x^3.

216 - 18.

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198

📝 Topic test — 8 questions

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Formula Reference Sheet

This chapter

Identities

Square of sum	(a + b)^2 = a^2 + 2ab + b^2
Difference of squares	a^2 - b^2 = (a + b)(a - b)
Cube of sum	(a + b)^3 = a^3 + b^3 + 3ab(a + b)
Sum/diff of cubes	a^3 +/- b^3 = (a +/- b)(a^2 -/+ ab + b^2)
Three-term	a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

x + 1/x family

Square	x^2 + 1/x^2 = (x + 1/x)^2 - 2
Cube	x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x)
Quadratic roots	x = (-b +/- root(b^2 - 4ac)) / 2a

SSC reference