For the equation `sqrt(mx-5)=x+3` ,the value of m is -3. What is the solution set for the equation?

Options

A. `{-3, 3}`
B. `{-2}`
C. `{-2, -7}`
D. `{3, 6}`

Answer

To solve the equation \( \sqrt{mx - 5} = x + 3 \) with \( m = -3 \), we will follow these steps:

### Step 1: Substitute the value of \( m \)
Given \( m = -3 \), we can substitute this value into the equation:
\[ \sqrt{-3x - 5} = x + 3 \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ (-3x - 5) = (x + 3)^2 \]

### Step 3: Expand the right side
Now, we expand the right side using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ -3x - 5 = x^2 + 6x + 9 \]

### Step 4: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ 0 = x^2 + 6x + 9 + 3x + 5 \]
This simplifies to:
\[ 0 = x^2 + 9x + 14 \]

### Step 5: Write in standard form
Rearranging gives us the standard quadratic equation:
\[ x^2 + 9x + 14 = 0 \]

### Step 6: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = 9 \)
- \( c = 14 \)

### Step 7: Use the quadratic formula
We apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 1 \cdot 14}}{2 \cdot 1} \]

### Step 8: Calculate the discriminant
Calculating the discriminant:
\[ 9^2 - 4 \cdot 1 \cdot 14 = 81 - 56 = 25 \]

### Step 9: Substitute back into the formula
Now substituting back into the quadratic formula:
\[ x = \frac{-9 \pm \sqrt{25}}{2} \]
\[ x = \frac{-9 \pm 5}{2} \]

### Step 10: Solve for \( x \)
We will now solve for the two possible values of \( x \):
1. For \( x = \frac{-9 + 5}{2} = \frac{-4}{2} = -2 \)
2. For \( x = \frac{-9 - 5}{2} = \frac{-14}{2} = -7 \)

### Final Solution Set
The solution set for the equation is:
\[ \{ -2, -7 \} \]

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