To find the solutions to the equation [tex]\(9x^2 - 54x = 0\)[/tex], let's go through a step-by-step process.
### Step 1: Identify the given quadratic equation
The given equation is:
[tex]\[ 9x^2 - 54x = 0 \][/tex]
### Step 2: Factor out the common term
First, we factor out the greatest common factor (GCF) from both terms. The GCF in this case is [tex]\(9x\)[/tex]. Therefore, we can factor the equation as:
[tex]\[ 9x(x - 6) = 0 \][/tex]
### Step 3: Apply the Zero-Product Property
The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ 9x = 0 \quad \text{or} \quad (x - 6) = 0 \][/tex]
### Step 4: Solve each equation separately
1. Solve [tex]\(9x = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
2. Solve [tex]\(x - 6 = 0\)[/tex]:
[tex]\[ x = 6 \][/tex]
### Step 5: List all possible solutions
The solutions to the equation [tex]\(9x^2 - 54x = 0\)[/tex] are:
[tex]\[ x = 0 \, \text{and} \, x = 6 \][/tex]
### Step 6: Match the solutions with the given options
From the given answer choices:
- A. [tex]\(x = 7\)[/tex]
- B. [tex]\(x = 6\)[/tex]
- C. [tex]\(x = 0\)[/tex]
- D. [tex]\(x = -7\)[/tex]
- E. [tex]\(x = -6\)[/tex]
We can see that the correct answers are:
- B. [tex]\(x = 6\)[/tex]
- C. [tex]\(x = 0\)[/tex]
Therefore, the solutions to the equation [tex]\(9x^2 - 54x = 0\)[/tex] are [tex]\(x = 6\)[/tex] and [tex]\(x = 0\)[/tex].
