Quadratic Equation with Two Real Roots — Finding the Sum
Answer
The answer is C — the sum of the roots is 7. For a quadratic ax² + bx + c = 0, the sum of the roots equals −b/a. Here −(−7)/1 = 7, so you never have to factor.
Problem
If x² − 7x + 10 = 0, what is the sum of the two solutions for x?
- A2
- B5
- C7✓ Correct
- D10
- E12
Step-by-step solution
Recall the shortcut. For ax² + bx + c = 0, the sum of the roots is −b/a and the product is c/a.
Read off the coefficients. Here a = 1, b = −7, c = 10.
Apply the formula. Sum of roots = −b/a = −(−7)/1 = 7. (Factoring confirms it: (x − 2)(x − 5) = 0, roots 2 and 5, sum 7.) The answer is C.
Key takeaway: When a quadratic only asks for the sum or product of roots, use −b/a and c/a instead of factoring — it is a one-step calculation.
Frequently asked
Do I need to factor to find the sum of the roots?
No. The sum of the roots of ax² + bx + c = 0 is always −b/a, so you can answer without finding the individual roots.
What is the product of the roots in this problem?
The product is c/a = 10/1 = 10, which matches 2 × 5.