A reliable method
Write a quadratic equation with zero on one side before applying the zero-product rule. If the question asks for area, dimensions must also satisfy the physical conditions.
The common trap
A negative algebraic root may be valid for the equation but invalid as a side length. Do not discard it without first distinguishing the equation from its context.
Check and transfer
Substitute each candidate into the original equation. For a vertex-form expression, inspect the squared term to determine when it is smallest and what value remains.
Worked comparison: why the alternative fails
The equation x(x+2)=24 becomes (x−4)(x+6)=0. Both 4 and −6 are algebraic roots, but only width 4 gives positive rectangle dimensions. A root can be invalid for a model without being invalid for the equation.
Transfer and explain
For h(x)=(x−2)²+9, the vertex input is 2 and minimum output is 9. Calculate h(0)=13 to contrast evaluation at zero with finding the minimum.
A rectangular garden has width x meters and length x + 3 meters. Its area is 28 square meters. What is its positive width?
- Write x(x + 3) equal to the stated area.
- Solve the quadratic and choose the positive length; the negative root is not a physical width.
- The requested value is 4. Check its units and meaning against the original question.
Compare the other answer choices
Report the sum of the two dimensions rather than the width.
Report area as width.
Treat the difference between dimensions as the actual length.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GED Mathematical Reasoning overviewSiyavula Grade 10 mathematics foundation reference