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Chapter 1: Problem 20

Find the zeros of the function algebraically. $$f(x)=9 x^{4}-25 x^{2}$$

Short Answer

Expert verified
The zeros of the function \(f(x) = 9x^{4} - 25x^{2}\) are: \(x = 0\), \(x=\frac{5}{3}\), and \(x=-\frac{5}{3}\).

Step by step solution

01

Set the Function Equal To Zero

To find the zeroes of the function, we need to find the values of \(x\) that make the function equal to zero. Thus our equation becomes \(9x^{4} - 25x^{2} = 0\).
02

Apply the Quadratic Formula

Applying the quadratic formula to the equation means treating \(x^{2}\) as the variable. Therefore,\(x^{2} = \frac{0 \pm \sqrt{0 - (4 * 9 * -25)}}{2 * 9}\).
03

Simplify the expression

Solving this gives us \(x^{2} = 0\) ,or \(x^{2} = \frac{25}{9}\). The first solution \(x^{2} = 0\) can only give us the solution \(x = 0\). However, solving the quadratic \(x^{2} = \frac{25}{9}\), we get \(x=\frac{5}{3}\), \(x=-\frac{5}{3}\).
04

State the result

Hence, the zeroes of function \(f(x) = 9x^{4} - 25x^{2}\) are: \(x = 0\), \(x=\frac{5}{3}\), and \(x=-\frac{5}{3}\).

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