Algebraic roots are candidates
Solving produces candidate values. The situation decides which candidates are meaningful and what final quantity must be reported.
Learning goals
- Form a quadratic model from a contextual situation.
- Solve or determine existence using factorisation or D.
- Filter roots and state a verified contextual answer.
Concept lesson
Model before solving
Define the unknown with units, express every related quantity in that variable, translate the condition, then standardise.
Choose solve or existence test
If dimensions or numbers are requested, factorise and find candidates. If the question asks whether a design is possible, D may settle real-solution existence before roots are needed.
Domain filter
Lengths and ages are positive; a digit is an integer 0-9 and a leading digit is non-zero; counts of objects are non-negative integers. Finish with the requested quantity and unit.
Prayer-hall dimensions
Let breadth be x m and length 2x+1 m. Area 300 m^2 gives x(2x+1)=300, hence 2x^2+x-300=0.
Solve and filter dimensions
2x^2+x-300=(2x+25)(x-12)=0 gives x=12 or -25/2. Reject the negative breadth; breadth is 12 m and length is 25 m. Check 12x25=300.
Filter every candidate
For each root pair, apply the supplied domain—length, age, digit or positive integer—and state why each rejected root is impossible.
Return to the question
Rewrite the requested result as a sentence with units and substitute it into every original condition, not only the quadratic equation.
Decide whether a grove is possible
Model the proposed length, breadth and area, calculate D and interpret whether real dimensions can exist. If real candidates exist, also apply positivity.
Application answer structure
Define, model, standardise, solve or test D, filter candidates, state the contextual answer with units and verify.
Exit ticket
The product of two consecutive positive integers is 306. Form and solve the equation, reject any invalid candidate and verify the integers in the original statement.