Solve the following equations: $\sqrt {x^{2} + ax - 1} - \sqrt {x^{2} + bx - 1} = \sqrt {a} - \sqrt {b}$.
- $-2, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 3}{(\sqrt {a} + \sqrt {b})^{2} - 3}$
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$-1, \dfrac {(\sqrt {a} - \sqrt {b})^{2} - 4}{(\sqrt {a} + \sqrt {b})^{2} + 4}$
- $\dfrac {1}{2}, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 4}{(\sqrt {a} + \sqrt {b})^{2} - 4}$
- $1, \dfrac {(\sqrt {a} - \sqrt {b})^{2} + 4}{(\sqrt {a} + \sqrt {b})^{2} - 4}$
Reveal answer
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D
Correct answer
Explanation
Let u = sqrt(x^2 + ax - 1) and v = sqrt(x^2 + bx - 1). The equation is u - v = sqrt(a) - sqrt(b). Squaring both sides and simplifying leads to the solution set including 1 and the expression in option D.
AI explanation
Assume x = 1 is a root by substitution, which satisfies the equation because both sides become zero. For the other root, after simplifying the equation by squaring and rearranging, we find it to be ((sqrt(a) - sqrt(b))^2 + 4) / ((sqrt(a) + sqrt(b))^2 - 4). Therefore, the roots are 1 and the given expression, which matches the correct option.