Answer :

Sure! Let's solve the equation algebraically step-by-step:

We have the equation:
[tex]\[ 5x^5 + 45x^3 = 0 \][/tex]

Our goal is to find all values of [tex]\( x \)[/tex] that satisfy this equation.

### Step 1: Factor Out the Greatest Common Factor

First, notice that both terms on the left side have a common factor of [tex]\( 5x^3 \)[/tex]. We can factor that out:

[tex]\[ 5x^3(x^2 + 9) = 0 \][/tex]

### Step 2: Set Each Factor to Zero

To find the solutions, we need to set each factor equal to zero:

1. [tex]\( 5x^3 = 0 \)[/tex]
2. [tex]\( x^2 + 9 = 0 \)[/tex]

### Step 3: Solve Each Equation

For [tex]\( 5x^3 = 0 \)[/tex]:

Divide each side by 5:

[tex]\[ x^3 = 0 \][/tex]

Take the cube root of both sides:

[tex]\[ x = 0 \][/tex]

For [tex]\( x^2 + 9 = 0 \)[/tex]:

Subtract 9 from both sides:

[tex]\[ x^2 = -9 \][/tex]

Take the square root of both sides:

[tex]\[ x = \pm \sqrt{-9} \][/tex]

Since we're dealing with real and imaginary numbers, we have:

[tex]\[ x = \pm 3i \][/tex]

Here, [tex]\( i \)[/tex] is the imaginary unit where [tex]\( i^2 = -1 \)[/tex].

### Step 4: List All Solutions

The solutions to the equation [tex]\( 5x^5 + 45x^3 = 0 \)[/tex] are:

[tex]\[ x = 0, \, x = 3i, \, x = -3i \][/tex]

These are all the values of [tex]\( x \)[/tex] that satisfy the original equation.

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