express p in terms of b if the quadratic equation f(x) = 2x power 2 +bx + 8 which is reflected at y=p has the same roots

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We have a quadratic function f(x) = 2x² + bx + 8. When this function is reflected across the horizontal line y = p, the resulting function has the same roots as the original. We need to find the value of p in terms of b. When we reflect a point across the line y equals p, the x-coordinate stays the same, but the y-coordinate becomes 2p minus y. For our function f(x), the reflected function g(x) equals 2p minus f(x). Substituting our original function, we get g(x) equals negative 2x squared minus bx plus 2p minus 8. For two quadratic equations to have the same roots, their coefficients must be proportional. We compare f(x) equals 2x squared plus bx plus 8, and g(x) equals negative 2x squared minus bx plus 2p minus 8. The ratio of leading coefficients is negative 1. The ratio of linear coefficients is also negative 1. For the constant terms, we need 8 divided by 2p minus 8 to equal negative 1. Now we solve for p. From our equation 8 divided by 2p minus 8 equals negative 1, we cross multiply to get 8 equals negative 1 times 2p minus 8. This simplifies to 8 equals negative 2p plus 8. Subtracting 8 from both sides gives us 0 equals negative 2p. Dividing by negative 2, we find that p equals 0. This means the reflection line is the x-axis. In conclusion, we found that p equals 0. This means the quadratic function f(x) equals 2x squared plus bx plus 8, when reflected across the x-axis, produces a function with the same roots. The answer is independent of the value of b, and p equals 0 for any quadratic of this form. This completes our solution.