Question

Solve the equation. Tell which solution method you used. \(-16 x^{3}+4 x=0\)

Short answer

The solutions for the equation \(-16x^3 + 4x = 0\) are \(x = 0, x = \frac{1}{2}, x = -\frac{1}{2}\). The method used to solve the equation is factorization.

Step-by-step solution

Factor Out Common Term

Notice, both terms in the equation \(-16x^3 + 4x = 0\) have a common factor of \(4x\). We factor out \(4x\) which simplifies the equation to \(4x (-4x^2 + 1) = 0\).

Identify the Roots

The equation will equal zero if either \(4x = 0\) or \(-4x^2 + 1 = 0\). By solving both, we get \(x = 0\) and \(-4x^2 = -1\), respectively.

Calculate X-values

We can now solve \(-4x^2 = -1\) for \(x\), yielding \(x = \sqrt{-1/-4}\) or \(x = -\sqrt{-1/-4}\). The final solutions for the equation are \(x = 0, x = \frac{1}{2}, x = -\frac{1}{2}\).