A reliable method
Move all terms to zero before using the zero-product property. Expand proposed factors to check them. A negative square plus a constant reveals a maximum directly.
The common trap
Discarding a negative root without a stated domain is wrong, but retaining a negative item count is also wrong. Return the requested root or combination, not an intermediate coefficient.
Check the conclusion
Substitute candidates into the original equation. Apply positive, integer and denominator restrictions after algebra, and check whether the requested value is attained.
Contrast the cases
The equation x²−2x=24 becomes (x−6)(x+4)=0, yielding 6 and −4. A positive-input condition removes −4; without that condition both roots remain. For P=100−(q−7)², the square is nonnegative, so P is at most 100 and reaches it at allowed input q=7. Here 7 is the maximizing input and 100 is the maximum output. Root-finding and optimization are different tasks.
Transfer and explain
Transfer: x²+x=30 has roots 5 and −6, sum −1 and product −30. Identify which answer serves each request, then state the positive root. Verify both roots in the original equation.
A model gives profit P = 300 − (q − 8)² for real q ≥ 0. What is the greatest possible profit?
- The squared term is at least zero.
- Profit is therefore at most the constant, and the bound occurs when the square is zero.
- At q = 8, the greatest profit is 300.
Compare the other answer choices
Confuses a root with the maximum value.
Returns the squared vertex input.
Evaluates at zero instead of the vertex.
At q=0, wrongly adds the square of the vertex input to the constant instead of subtracting it, then treats this as a maximum.
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.
GMAT exam contentSiyavula Grade 10 mathematics foundation reference