Choose a method
Choose the form that exposes the requested feature. Expand factors to verify an equivalence, and substitute a proposed zero into the original expression.
Avoid this trap
The sign inside x−r corresponds to a zero at positive r. The coordinate of a vertex is not automatically an x-intercept.
Check and transfer
A context can restrict solutions to positive lengths or nonnegative times. Apply that restriction only after finding the algebraic candidates.
Worked example with a numerical check
The function q(x) = (x − 4)(x + 6) has two zeros. What is their distance on the number line? Set each factor equal to zero to obtain the two roots. Subtract the negative root from the positive root to get their distance. The requested value is 10.
Contrast and diagnose
The function h(x) = 2(x − 4)² + 6 has a minimum. At what x-value does the minimum occur? A square is nonnegative, and multiplying by two preserves that. The minimum occurs when the squared expression is zero. The requested value is 4. An incorrect approach: Multiply the horizontal coordinate by the vertical stretch.
Transfer practice
If (x + 4)² = x² + bx + 16 for every real x, what is b? Try first. Check: 8. Expand the square as a product of two identical binomials. Add the two cross terms to obtain the coefficient of x.
The function q(x) = (x − 4)(x + 6) has two zeros. What is their distance on the number line?
- Set each factor equal to zero to obtain the two roots.
- Subtract the negative root from the positive root to get their distance.
- The requested value is 10.
Compare the other answer choices
Report the magnitude of the root sum instead of the separation.
Report only one root’s magnitude.
Subtract constants without accounting for the negative root.
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.
Math specifications