If one root of the quadratic equation ax2 + bx + c = 0 is the reciprocal of…

Concept

For a quadratic equation ax2 + bx + c = 0 with roots α and β, the standard root-coefficient relations give: sum of roots = -b/a, and product of roots = c/a.

Application

1. One root is the reciprocal of the other, so let the roots be α and 1/α.

2. The product of these two roots is α × (1/α) = 1.

3. By the root-coefficient relation, the product of the roots also equals c/a.

4. Equating the two expressions for the product: c/a = 1.

5. Solving this equation gives c = a, i.e. a = c.

Cross-check

Take a = c = 1 and b = -5, giving x2 - 5x + 1 = 0. Its two roots multiply to c/a = 1/1 = 1, so each root is exactly the reciprocal of the other, confirming that a = c satisfies the reciprocal-root condition independently of the value of b.