Choosing algebra first means using a relationship between the quantities instead of finding each value separately. It appears whenever a question gives a sum, a product or a reciprocal and asks for a square or a combination, and it is one of the main skills in advanced non-calculator reasoning.
When is the shorter route available?
The shorter route exists when the required expression is symmetrical in the unknowns. Expressions like a² + b², (a − b)² and x² + 1/x² can all be written using a + b and ab, or using x + 1/x.
The central identity is (a + b)² = a² + 2ab + b². Rearranging gives a² + b² = (a + b)² − 2ab.
For a reciprocal pair, treat b as 1/x. Then ab = 1, so x² + 1/x² = (x + 1/x)² − 2.
Worked example
Given that x + 1/x = 5, find the value of x² + 1/x².
Step 1, choose the structure: the target is a square of each term, and the given is their sum.
Step 2, square the given: (x + 1/x)² = x² + 2 × x × (1/x) + 1/x² = x² + 2 + 1/x².
Step 3, substitute the number: 5² = x² + 2 + 1/x², so 25 = x² + 2 + 1/x².
Step 4, rearrange: x² + 1/x² = 25 − 2 = 23.
Check by the long way: x + 1/x = 5 means x² − 5x + 1 = 0, so x = (5 + √21)/2 ≈ 4.791. Then x² ≈ 22.956 and 1/x² ≈ 0.044, and the sum is 23.000 to three decimal places. Both routes agree.
Notice that the long route needed a quadratic, a surd and two squarings, while the algebra took four lines with no surds.
The mistake to watch for
A common slip is to drop the middle term when squaring.
Mistaken working: (x + 1/x)² = x² + 1/x², so x² + 1/x² = 25
The student squared each term and forgot the cross term 2 × x × (1/x) = 2.
The correction is to write out the full square every time, and to notice that the cross term here equals 2 because x × (1/x) = 1. Then subtract it.
Check yourself
Try these without a calculator, then open each answer.
1. Given that a + b = 7 and ab = 10, find a² + b².
Show answer
a² + b² = (a + b)² − 2ab = 49 − 20 = 29. Check: a = 5, b = 2 gives 25 + 4 = 29.
2. Given that x − 1/x = 3, find x² + 1/x².
Show answer
(x − 1/x)² = x² − 2 + 1/x² = 9, so x² + 1/x² = 9 + 2 = 11. The middle term is now negative, so you add 2.
3. Without a calculator, find 101² − 99².
Show answer
Difference of squares: (101 − 99)(101 + 99) = 2 × 200 = 400. Check: 10 201 − 9 801 = 400.
Where this leads next
After this, try simplifying a long expression by recognising structure. The quadratic structure explorer shows how a sum and product of roots link to the coefficients, which is the same idea in another form. The non-calculator working trainer gives you extra exact-arithmetic practice.
Some students understand the identity but do not think to use it when a question looks unfamiliar. That habit of asking “what is the question really about?” is something our teachers build in online one-to-one Additional Mathematics tuition.