Lens 02 of 9
The landmark
Look at the number the answer must sit near, or the easy numbers either side of it. Test outward from there.
A total, a product or a square root tells you where the answer lives before it tells you what it is. Two squares adding to 394 means each is about 197, and √197 is just under 14. Find that landmark first; then the answer is one or two checks away, not a quadratic away.
The clues that call for it
- Quadratic Equations
The sum of two squares is given
394 ÷ 2 = 197 and √197 ≈ 14, so look either side of 14.
- Quadratic Equations
A product of two close numbers is given
√224 ≈ 15, so the consecutive evens are 14 and 16.
- Statistics
n/2 in a table of running totals
n = 40: the median sits where the running totals first pass 20.
- Anywhere
A root or a size between two easy numbers
44² = 1936 and 45² = 2025 trap √2000 between 44 and 45.
Worked example
Two consecutive odd integers have squares summing to 394. Find them.
- 1
Drop an anchor
If the two squares are close, each is about half the total.
394 ÷ 2 = 197
- 2
Find the landmark
√197 is just under 14, so the numbers sit either side of 14.
√197 ≈ 14
- 3
Test under the parity condition
The odd integers straddling 14 are 13 and 15.
- 13
- 15
- 4
Verify
169 + 225 = 394 ✓. The landmark turned 'solve a quadratic' into 'check two numbers'.
Try the lens
Two consecutive even integers have squares summing to 340. What is the larger one?
Problems that use this lens
- 1Two even neighboursFind two consecutive positive even integers whose product is 224.Lens · The landmarkquadratic equationsPractice (NCERT style)
- 2Two odd squaresThe sum of the squares of two consecutive odd positive integers is 394. Find them.Lens · The landmarkquadratic equationsJEE-style practice
- 3Two apart, squares 202Two natural numbers differ by 2 and the sum of their squares is 202. The sum of the numbers is:Lens · The landmarkquadratic equationsPractice
- 4Consecutive squares, 365The sum of the squares of two consecutive positive integers is 365. The sum of the numbers is:Lens · The landmarkquadratic equationsNCERT Ex 4.2 Q4, as a multiple-choice question
- 5Three apart, product 504Two numbers differ by 3 and their product is 504. Find the numbers.Lens · The landmarkquadratic equationsPractice
- 1Finding the median classThe marks of 40 students are grouped: 0–10: 4, 10–20: 14, 20–30: 8, 30–40: 9, 40–50: 5. Find the median mark.Lens · The landmarkstatisticsPractice (NCERT style)