If `3x^(2)+9x=84`, what are the possible values of x ?

Options

A. `-4 and 7`
B. `-7 and 4`
C. `-7 and -4`
D. `-7 and -12`

Answer

To solve the equation \(3x^2 + 9x = 84\), we will follow these steps:

### Step 1: Rearrange the equation
First, we will move all terms to one side of the equation to set it to zero:
\[ 3x^2 + 9x - 84 = 0 \]

### Step 2: Simplify the equation
Next, we can simplify the equation by dividing all terms by 3:
\[ x^2 + 3x - 28 = 0 \]

### Step 3: Identify coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), we identify:
- \(a = 1\)
- \(b = 3\)
- \(c = -28\)

### Step 4: Use the quadratic formula
We will use the quadratic formula to find the roots of the equation:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-28)}}{2 \cdot 1} \]

### Step 5: Calculate the discriminant
Now, we calculate the discriminant:
\[ b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot (-28) = 9 + 112 = 121 \]

### Step 6: Substitute back into the formula
Now we substitute the discriminant back into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{121}}{2} \]

### Step 7: Simplify the square root
Since \(\sqrt{121} = 11\), we have:
\[ x = \frac{-3 \pm 11}{2} \]

### Step 8: Calculate the two possible values of \(x\)
Now we can calculate the two possible values:
1. For \(x = \frac{-3 + 11}{2}\):
\[ x = \frac{8}{2} = 4 \]
2. For \(x = \frac{-3 - 11}{2}\):
\[ x = \frac{-14}{2} = -7 \]

### Final Answer
The possible values of \(x\) are:
\[ x = 4 \quad \text{and} \quad x = -7 \]

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