Question

one of the solutions to the equation $x^3 + 5x^2 + 10x + 50 = 0$ is $-5$. find the full solution set. which of the following options correctly describes the solution set? option #1: it has one real solution and two non - real solutions. option #2: it has one non - real solution and two real solutions. option #3: it has three real solutions. (1 point) option #\square

Explanation:

Step1: Factor the cubic equation

Given the equation \(x^3 + 5x^2 + 10x + 50 = 0\), we can factor by grouping. Group the first two terms and the last two terms:
\((x^3 + 5x^2) + (10x + 50) = 0\)
Factor out the greatest common factor from each group:
\(x^2(x + 5) + 10(x + 5) = 0\)
Now, factor out \((x + 5)\) from both terms:
\((x + 5)(x^2 + 10) = 0\)

Step2: Solve for \(x\)

Set each factor equal to zero:

1. For \(x + 5 = 0\), we get \(x = -5\) (this is the given real solution).
2. For \(x^2 + 10 = 0\), we have \(x^2 = -10\). Taking the square root of both sides, we get \(x = \pm\sqrt{-10} = \pm i\sqrt{10}\) (these are non - real, imaginary solutions).

So, the equation has one real solution (\(x=-5\)) and two non - real solutions (\(x = i\sqrt{10}\) and \(x=-i\sqrt{10}\)).

Answer:

Option #1: It has one real solution and two non - real solutions.