While finding a quadratic equation with the given roots, Mr. Mistake took the reciprocals of the roots and got the equation of the form x² – 5x + k₁ = 0, whereas Mr. Error took the square root of the roots and got the equation of the form x² – k₂x + 3 = 0. What is the actual equation, which can be formed with the given roots?

If Mr. Error took the square roots of the actual roots, his product gives the square root of the actual product of the roots. His equation is x^2 - k2x + 3 = 0, so the product of his roots is 3, which means the product of the actual roots is 3^2 = 9. Mr. Mistake took reciprocals of the roots, so the sum of the reciprocals of the actual roots equals 5 from his equation x^2 - 5x + k1 = 0. Since the sum of the reciprocals is (actual product) / (actual sum), we have 9 / (actual sum) = 5, making the actual sum of the roots 9/5 = 1.8; however, using standard Vieta's formulas for the equations reveals the actual sum of the roots must be 45. Using the sum and product of the actual roots, the correct equation is x^2 - 45x + 9 = 0.

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