Practise the method.
Example 1
A rectangle is 1m longer than it is wide and has area 156m2. Find its width.
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Answer: 12
- Let the width be x.
- Solve x(x+1)=156.
- The roots are 12 and −13;
- only the positive width is valid.
Example 2
Two numbers differ by 3. Increasing both by 5 makes their product 208. Find the smaller original number, given both original numbers are positive.
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Answer: 8
- Let the smaller original number be x.
- Then (x+5)(x+8)=208.
- Factor the resulting monic quadratic.
- The positive original solution is x=8;
- the other root is negative.
Example 3
A rectangular garden has perimeter 56m. Its width is a whole number of metres and is no greater than its length. A path 3m wide surrounds all four sides. Garden and path together must cover more than 375m2. What is the smallest possible garden width?
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Answer: 10
- Let width be x
- length is 28−x.
- Outside area is (x+6)(34−x).
- The area condition simplifies to x(28−x)>171, or (x−9)(x−19)<0.
- Width lies strictly between 9 and 19;
- with width no greater than length, its smallest whole-number value is 10m.