The expression x+x1 (and its minus twin x−x1) is the single most-asked algebra shape in SSC. The trick is that x and x1 multiply to 1, so the middle term of every expansion is just a plain number — and a tidy ladder appears.
The reciprocal ladder
Start from x+x1=k. Squaring gives (x+x1)2=x2+x21+2 (the middle term 2⋅x⋅x1=2), so:
x2+x21=k2−2 and x3+x31=k3−3k.
The minus version flips the small signs. From x−x1=k:
x2+x21=k2+2 and x3−x31=k3+3k.
Anchor the trap: a plus given subtracts 2, a minus given adds 2. If you are handed x2+x21 instead, recover k first via k=(x2+x21)+2, then climb.
🔷Identity ExplorerPick an identity to see its proof
(a+b)² = a² + 2ab + b²
Expand as (a+b)×(a+b). The square area splits into 4 rectangles: a², ab, ab, b².
Numeric Verification
LHS: (3+2)² = 25
RHS: 3²+2·3·2+2² = 25
✓ LHS = RHS — identity verified!
If x+x1=5, then x2+x21=52−2= .
📝Practice Questions
Q1If x + 1/x = 4, find x² + 1/x².
Q2If x − 1/x = 3, find x² + 1/x².
Q3If a + 1/a = 4, find a³ + 1/a³.
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Real exam questions — Reciprocal Ladders
2 question types · 7 solved examples from real SSC papers
The reciprocal ladder (x±x1 and its powers) is the single most-asked algebra shape in SSC. Two formulas per direction cover every variant.
How to solve this type
Two formulas do it all. Square: x2+x21=k2−2 (the −2 comes from the middle term 2⋅x⋅x1). Cube: x3+x31=k3−3k. If you are instead given x2+x21 and want the cube, first recover k=(x2+x21)+2, then cube.
