If α and β are the roots of ax^2 + bx + c = 0, then α + β equals what?

If α and β are the roots of ax^2 + bx + c = 0, then α + β equals what?

• A. -b/a
• B. -a/b
• C. b/a
• D. a/b

Answer: (A)

In a quadratic, the sum of the roots is tied to the linear coefficient. Write the quadratic with roots α and β as a(x − α)(x − β). Expanding gives ax^2 − a(α + β)x + aαβ. Comparing the coefficient of x with ax^2 + bx + c shows that −a(α + β) = b, so α + β = −b/a (and αβ = c/a as a consistency check). This is why the sum of the roots is −b divided by a. Any other expression would not align with the coefficient matching when you expand back to the original form.

In a quadratic, the sum of the roots is tied to the linear coefficient. Write the quadratic with roots α and β as a(x − α)(x − β). Expanding gives ax^2 − a(α + β)x + aαβ. Comparing the coefficient of x with ax^2 + bx + c shows that −a(α + β) = b, so α + β = −b/a (and αβ = c/a as a consistency check). This is why the sum of the roots is −b divided by a. Any other expression would not align with the coefficient matching when you expand back to the original form.