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Chapter 1: Problem 47

Find the value(s) of \(x\) for which \(f(x)=g(x)\). $$f(x)=x^{4}-2 x^{2}, \quad g(x)=2 x^{2}$$

Short Answer

Expert verified
The values of \(x\) for which \(f(x) = g(x)\) are \(x = 0, -2, 2\).

Step by step solution

01

Set Up The Equation

Start by setting \(f(x)\) equal to \(g(x)\): \(x^{4}-2 x^{2} = 2 x^{2}\).
02

Rearrange The Equation

Rearrange the above equation to combine like terms: \(x^{4}- 4x^{2} = 0\)
03

Factor Out The Common Term

Factor out the common term (\(x^{2}\)) to make it easier to solve for \(x\): \(x^{2}(x^{2}-4) = 0\)
04

Apply Zero Product Property

Use the zero product property which states that if the product of two factors is zero, then at least one of the factors must be zero: \(x^{2} = 0 \) or \(x^{2}-4 = 0 \)
05

Solve for x

Solving the two equations from the previous step yields: \(x = 0, -2, 2\)

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