Which of the following quadratic equations has the sum of its roots equal to 11?

Introduction / Context:This question checks your understanding of the relationship between the coefficients of a quadratic equation and the sum of its roots. For a quadratic equation of the form ax^2 + bx + c = 0 with a ≠ 0, the sum of the roots is given by −b/a. You must apply this simple formula to each option and identify which one yields a sum of 11.

Given Data / Assumptions:

• We are given several quadratic equations in x with real coefficients.
• For each equation ax^2 + bx + c = 0, the sum of the roots is −b/a.
• We need the equation whose roots add up to 11.
• All equations have leading coefficient a = 1, which simplifies calculations.

Concept / Approach:For a quadratic equation ax^2 + bx + c = 0 with roots α and β, Viète's formulas state that α + β = −b/a and αβ = c/a. Because a = 1 in each option, the sum of roots is simply −b. Therefore, by inspecting the coefficient of x in each equation, we can quickly find the sum of its roots without solving the equation explicitly. We then compare this sum with the required value 11.

Step-by-Step Solution:Option a: x^2 − 11x + 18 = 0 has coefficients a = 1 and b = −11, so the sum of its roots is −b/a = −(−11)/1 = 11.Option b: x^2 − 7x + 10 = 0 has b = −7, so the sum of roots is 7, not 11.Option c: x^2 + 2x − 26 = 0 has b = 2, so the sum of roots is −2.Option d: x^2 + 5x − 6 = 0 has b = 5, so the sum of roots is −5.Option e: x^2 − 13x + 36 = 0 has b = −13, so the sum of roots is 13.Only option a yields the required sum of 11.

Verification / Alternative check:We can also factor option a to verify more concretely. The equation x^2 − 11x + 18 = 0 factors into (x − 2)(x − 9) = 0. The roots are x = 2 and x = 9, and their sum is 2 + 9 = 11, exactly as desired. This confirms that option a is correct. By contrast, factoring option b gives roots 2 and 5, whose sum is 7; option e factors to roots 4 and 9, whose sum is 13, not 11.

Why Other Options Are Wrong:

• Option b has a sum of roots equal to 7, which is too small.
• Option c has a sum of roots equal to −2, which has the wrong sign as well as magnitude.
• Option d has a sum of roots equal to −5.
• Option e has a sum of roots equal to 13, which is close but not equal to 11.

Common Pitfalls:

• Misremembering the sign in the formula and writing the sum as b/a instead of −b/a.
• Attempting to solve each quadratic by factoring or using the quadratic formula, which is slower than using Viète's formula.
• Confusing the sum of roots with the product of roots c/a.

Final Answer:x^2 - 11x + 18 = 0