x + 1/x Problems: From x + 1/x to x² + 1/x², x³ + 1/x³ and Beyond

SSC papers have a family of questions that start "if x + 1/x = k" and ask for x² + 1/x², x³ + 1/x³ or something higher. You never solve for x. Instead, you apply the square or cube identities to the whole expression x + 1/x. The product of x and 1/x is always 1, which is why these questions come out so neatly.

The same few rules handle the minus versions, x − 1/x, and questions disguised as quadratic equations. Learn the four formulas below and every question in this family takes one or two lines.

Each comes from expanding a bracket. (x + 1/x)² = x² + 2 + 1/x², because the middle term is 2 × x × 1/x = 2. (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x), because the middle terms are 3 × x × 1/x times (x + 1/x).

And x³ − 1/x³ = 27 + 9 = 36.

For fourth powers, apply the square rule twice. Treat x² as the new variable: x⁴ + 1/x⁴ = (x² + 1/x²)² − 2.

For a sixth power, either square x³ + 1/x³ and subtract 2, or cube x² + 1/x² and subtract three times it. Both routes give the same answer, which makes a handy check.

Going backwards is also common. Given x² + 1/x², add 2 and take the square root to recover x + 1/x.

An equation of the form x² − kx + 1 = 0 is an x + 1/x question. Divide every term by x to get x − k + 1/x = 0, so x + 1/x = k.

Two special values make every power easy. If x + 1/x = 2, then x = 1, so xⁿ + 1/xⁿ = 2 for every n. If x + 1/x = −2, then x = −1, and the answer alternates between −2 for odd n and 2 for even n.

• ✘ Subtracting 2 for x − 1/x. With the minus version, add 2.
• ✘ Cubing without the −3a term: x³ + 1/x³ is not simply a³.
• ✘ Forgetting to divide by x when the question is a quadratic.
• ✘ Taking only the positive square root when the sign of x is not given.

The general versions of these rules are on the algebraic identities page. For quadratic equations in bank exams, see comparing roots of quadratics, and the formula list is in the Quant cheat code.

Can x + 1/x be any number?

For real x, x + 1/x is never between −2 and 2. It equals 2 only at x = 1 and −2 only at x = −1. If a question gives a value such as 1 or √3, x is not real, but the identities still give the right answer for the expression asked.

How do I find x + 1/x from x² + 1/x²?

Add 2 and take the square root: (x + 1/x)² = x² + 1/x² + 2. If the question says x is positive, take the positive root. Otherwise both signs are possible, and the answer may come out as plus or minus.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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