Build, solve and evaluate a quadratic model
Define a variable with units and a domain. Translate the relationship, solve the quadratic and check the original situation. A mathematical root can satisfy the equation without describing a possible measurement
- Choose the relationship Rectangle area is a product; triangle area is half the product of base and perpendicular height
- Translate dimensions An outside border adds twice its width to each dimension. An inside path subtracts twice its width
- Specify a domain All dimensions must be positive; counts may need to be integers; time is restricted to the modelled interval
- Solve and interpret Put zero on one side, factorise and solve both branches before applying the domain
- Evaluate and report Substitute into the original situation, state assumptions and revise the model if it does not fit the observations
Worked example
A rectangular inner card has sides x and x+2 cm. A uniform 1 cm outside border gives total outer area 48 cm². Find the inner dimensions
- \(\displaystyle L=x+4,\quad W=x+2\)Add the border at both ends of each inner dimension
- \(\displaystyle (x+4)(x+2)=48\)Use the outer area, including the card and border
- \(\displaystyle x^2+6x-40=0\)Expand and subtract 48.
- \(\displaystyle (x+10)(x-4)=0\)The factor constants multiply to −40 and add to 6.
- \(\displaystyle x=-10\text{ or }x=4\)Only 4 gives positive inner dimensions
- \(\displaystyle 6\times 8=48\)The inner dimensions are 4 cm by 6 cm; the outer dimensions are 6 cm by 8 cm. This checks the original total area
Common mistakes
Difficulty levels
Questions about this topic
Does a quadratic always give two usable answers?+
No. Check every mathematical root against the original restrictions. Negative dimensions, non-integer counts or times outside the model may be excluded
Why does a border add twice its width?+
Each full dimension crosses a strip at both ends. An inside path instead removes a strip at each end
Which triangle height should I use?+
The perpendicular height to the chosen base, not an arbitrary sloping side
Can equivalent equations have different coefficients?+
Yes. Multiplying a zero-form equation by a nonzero constant preserves its solutions. Coefficient tables therefore state the required normalisation
Why is the height coefficient negative five?+
This idealised vertical-motion model uses gravity approximately 10 metres per second squared and ignores air resistance. The coefficient is half the downward acceleration
What if a measurement disagrees with the model?+
Check the measurements, units and calculations, then reassess the assumptions and modify the model where needed
What should a final report contain?+
The variable and units, domain, relationship, solution, original-context check and relevant assumptions or limits
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs