Build and solve factorisable quadratic models, then interpret their solutions.
Free examples and worked answers. No account needed.
Practise the method.
Build and solve factorisable quadratic models, then interpret their solutions.
Build and solve factorisable quadratic models, then interpret their solutions.
Example 1
A rectangle is 1m longer than it is wide and has area 156m2. Find its width.
RectangleDiagram not to scaleShow worked answer
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.
Show worked answer
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?
Garden and surrounding pathDiagram not to scaleShow worked answer
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.
Put it into practice.
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