Stories that square
A word problem becomes an equation once the unknown gets a letter. The unknown is the value the story asks for.
Step 1 of 4
- A guava orchard: length 3 m more than the breadth.
- Let the breadth be
b. Length isb + 3. - Area = length × breadth = 40 m². So
b(b + 3) = 40. - Tidied:
b² + 3b − 40 = 0.
The rule
- Name the unknown, with its unit.
- Write one equation from the story.
- Solve it for real roots.
- Check each root against the story.
The orchard's decider is 9 + 160 = 169. Its square root is 13. The roots are (−3 ± 13)/2. That gives 10/2 = 5 and −16/2 = −8. A breadth of −8 m is not possible. So the breadth is 5 m. The length is 5 + 3 = 8 m.
Numbers in a row follow each other, like 19 and 20.
| Story | Equation | Roots | The story keeps |
|---|---|---|---|
| Orchard, breadth b | b² + 3b − 40 = 0 | 5, −8 | breadth 5 m |
| Plot 3 m shorter than twice breadth, area 54 m² | 2b² − 3b − 54 = 0 | 6, −9/2 | breadth 6 m |
| Two positive whole numbers in a row, first n, product 380 | n² + n − 380 = 0 | 19, −20 | numbers 19 and 20 |
These three equations each have two real roots. A story often keeps one. A length, a speed or a count cannot be negative.
Now try it. Pick a story. Slide the unknown until it balances.
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