Question

Solve the equation. Tell which solution method you used. \(5 x^{4}-80 x^{2}=0\)

Short answer

The solutions to the quadratic equation \(5 x^{4}-80 x^{2}=0\) are \(x=-2, 2, -2i\), and \(2i\). The solution method used was factoring.

Step-by-step solution

Rewrite the Equation

To simplify the equation, first, rewrite it in the form of a basic quadratic equation. Recast the given formula \(5 x^{4}-80 x^{2}=0\) as \(5y^2 - 80 = 0\), where \(y = x^2\)

Factorize the Equation

Then, factorize the quadratic equation, which is \(5y^2 - 80 = 0\), to find the roots of the equation. So, it can be rewritten as \(5(y^2-16)=0\), then further factorized into \(5(y-4)(y+4)=0\)

Solve for y

At this point, we can solve for the variable \(y\). The solutions to the equation are found by setting each of the factors equal to zero and solving for \(y\), giving \(y=-4\) or \(y=4\)

Revert y to x and Solve

As our original equation is in terms of variable \(x\), revert back the \(y\) to \(x^2\). Solving \(x^2=4\) and \(x^2=-4\), we find \(x=-2, 2, -2i, 2i\)

Checking the Solutions

Lastly, we should check these solutions by plugging them back into the original equation for \(x\) to see if they make the equation true. In this case, all the solutions are valid.