To solve the equation [tex]\(4x(x + 3) = 0\)[/tex], let's go through the steps in a detailed manner.
### Step 1: Understand the Structure of the Equation
The given equation is a product of two expressions set to zero:
[tex]\[ 4x(x + 3) = 0 \][/tex]
This is a quadratic equation that can be factored into two simpler factors, [tex]\(4x\)[/tex] and [tex]\(x + 3\)[/tex].
### Step 2: Set Each Factor to Zero
Since the product of two factors is zero, at least one of the factors must be zero. We can set each factor to zero and solve for [tex]\(x\)[/tex].
#### Factor 1: [tex]\(4x = 0\)[/tex]
Divide both sides by 4:
[tex]\[ x = 0 \][/tex]
#### Factor 2: [tex]\(x + 3 = 0\)[/tex]
Subtract 3 from both sides:
[tex]\[ x = -3 \][/tex]
### Step 3: Verify the Solutions
Let's check if both solutions satisfy the original equation:
1. For [tex]\(x = 0\)[/tex]:
[tex]\[ 4(0)(0 + 3) = 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
This is true.
2. For [tex]\(x = -3\)[/tex]:
[tex]\[ 4(-3)(-3 + 3) = 0 \][/tex]
[tex]\[ 4(-3)(0) = 0 \][/tex]
[tex]\[ 0 = 0 \][/tex]
This is also true.
### Step 4: Summary of Solutions
So, the solutions to the equation [tex]\(4x(x + 3) = 0\)[/tex] are:
[tex]\[ x = 0 \][/tex]
and
[tex]\[ x = -3 \][/tex]
Therefore, the complete set of solutions is:
[tex]\[ x = 0 \; \text{and} \; x = -3 \][/tex]
