Question

Solve the equation by factoring, by finding square roots, or by using the quadratic formula. $$4 n^{2}+2 n=0$$

Short answer

The solutions to the equation are \(n = 0\) and \(n = -1/2\).

Step-by-step solution

Simplify the Equation

First, simplify the quadratic equation by factoring out the greatest common factor, in this case, it's \(2n\). So, the equation becomes: \(2n(2n+1)=0 \)

Use Zero Product Property

Next, apply the Zero-Product Property, which means if a product of two factors is zero, then at least one of the factors must be zero. In this case, we can set both \(2n\) and \((2n+1)\) equal to zero and solve for \(n\).

Solve for n

Setting \(2n = 0\) simplifies to \(n = 0\). And setting \(2n + 1 = 0\) simplifies to \(n = -1/2\).