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Answer :
Final answer:
To find the zeros of the function f(x) = 9x^4 - 250x^2 + 625, we set the function equal to zero and use a substitution to reduce it to a quadratic equation. We then apply the quadratic formula to find the solutions for the substituted variable and revert the substitution to determine the zeros of x.
Explanation:
Finding Zeros of a Quartic Function The question involves finding the zeros of the function f(x) = 9x4 - 250x2 + 625. To find the zeros of the function, we need to set the function equal to zero and solve for x. Observing the structure of the equation, we see it resembles a quadratic in form, with x2 as the variable. We can let u = x2 and subsequently solve the resulting quadratic equation 9u2 - 250u + 625 = 0 for u using the quadratic formula. To apply the quadratic formula, we identify the coefficients a = 9, b = -250, and c = 625. The quadratic formula is given by x = (-b ± sqrt(b2 - 4ac)) / (2a). Substituting the coefficients, we can find the values for u. After finding the solutions for u, we take the square root of each to find the values of x, thus determining the zeros of the original function. To check your solutions, evaluate f(x) at each value of x you find; if the function equals zero, you have confirmed that the corresponding value of x is indeed a zero of the function. Keep in mind that when taking the square root of both sides to solve for x, both the positive and negative roots need to be considered unless there is a constraint that defines x within a certain interval.
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