Let's solve the equation [tex]\(6x - 3x^{\frac{3}{2}} = 0\)[/tex] step by step.
### Step 1: Factor out the common terms
First, observe that both terms on the left side of the equation have a common factor of [tex]\(3x\)[/tex]. We can factor out this common term:
[tex]\[6x - 3x^{\frac{3}{2}} = 3x(2 - x^{\frac{1}{2}}) = 0\][/tex]
### Step 2: Set each factor to zero
Next, we apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
So, set each factor equal to zero:
1. [tex]\(3x = 0\)[/tex]
2. [tex]\(2 - x^{\frac{1}{2}} = 0\)[/tex]
### Step 3: Solve each equation separately
Now, we solve these simpler equations one by one.
#### For [tex]\(3x = 0\)[/tex]:
Divide both sides by 3:
[tex]\[x = 0\][/tex]
#### For [tex]\(2 - x^{\frac{1}{2}} = 0\)[/tex]:
Add [tex]\(x^{\frac{1}{2}}\)[/tex] to both sides:
[tex]\[2 = x^{\frac{1}{2}}\][/tex]
Square both sides to get rid of the square root:
[tex]\[(2)^2 = \left(x^{\frac{1}{2}}\right)^2\][/tex]
[tex]\[4 = x\][/tex]
### Step 4: Write down all solutions
The solutions we have found are:
[tex]\[x = 0\][/tex]
[tex]\[x = 4\][/tex]
Therefore, the complete solution set for the equation [tex]\(6x - 3x^{\frac{3}{2}} = 0\)[/tex] is:
[tex]\[x = 0\][/tex]
[tex]\[x = 4\][/tex]
In conclusion, the values of [tex]\(x\)[/tex] that satisfy the given equation are [tex]\(x = 0\)[/tex] and [tex]\(x = 4\)[/tex].
