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Let a, b, c be real numbers with a ≠ 0. Suppose α is a solution of a x² + b x + c = 0 and β is a solution of a x² - b x - c = 0, with 0 < α < β. For the quadratic equation having α and β as its roots, which of the following is always a root?

1. γ = (α + β)/2
2. γ = (α - β)/2
3. γ = α
4. α < γ < β

Correct answer: γ = α

Solution

The correct option is that b3 = b1 because b1 is one of the roots of the quadratic equation formed by b1 and b2. Since b1 is a root, it must satisfy the equation, making it a valid solution.

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