If the arithmetic mean of the roots of a quadratic equation is $\dfrac{8}{5}$ and the arithmetic mean of their reciprocals is $\dfrac{8}{7}$, then the equation is

Let the roots of the quadratic equation be r and s. Using the sum and product of roots formulas, their arithmetic mean is (r + s)/2 = 8/5, so the sum r + s = 16/5. The arithmetic mean of their reciprocals is (1/r + 1/s)/2 = 8/7, which simplifies to (r + s)/(2rs) = 8/7. Substituting the sum gives (16/5) / (2rs) = 8/7, which solves to rs = 14/5. A quadratic equation is given by x^2 - (sum)x + (product) = 0, yielding x^2 - 16/5 x + 14/5 = 0. Multiplying the entire equation by 5 results in 5x^2 - 16x + 7 = 0.

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