A reliable method
Set the equation equal to zero. For ax² + bx + c, examine b² − 4ac before deciding whether there are two, one or no distinct real solutions.
The common trap
A zero discriminant gives one distinct real root with multiplicity two. It does not mean the root itself is zero.
Check and transfer
The roots sum to −b/a and multiply to c/a. Use these relationships when the requested quantity is a sum or product rather than solving for every root separately.
Worked comparison: identify the structure
For 2x²−6x+4=0, roots 1 and 2 have sum 3 and product 2. The coefficient shortcuts −b/a and c/a reproduce those quantities. Omitting the leading coefficient gives 6 and 4, neither the requested sum nor product.
Transfer and explain
For x²−8x+c=0 to have a repeated real root, set 64−4c=0, so c=16. The repeated root is 4, illustrating why c and the root itself must not be confused.
What is the sum of the roots of 3x² − 18x + 2 = 0?
- For ax² + bx + c = 0, the root sum is −b/a.
- Use the signed coefficient b rather than only its magnitude.
- The requested value is 6. Substitute or reverse the operation to check this result.
Compare the other answer choices
Divide by the leading coefficient twice.
Report the root product c/a instead of the sum.
Use b/a instead of negative b/a.
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.
CLEP College Algebra exam overviewSiyavula Grade 10 foundation reference