If `f(x)=3x^(2)+24x-53 ` , find the negative value of `f^(-1)(0)`.

Options

A. `-58.80`
B. `-9.80`
C. `-8.23`
D. `-1.87`

Explanation

To find the negative value of \( f^{-1}(0) \) for the function \( f(x) = 3x^2 + 24x - 53 \), we need to follow these steps:

### Step 1: Set the function equal to zero
Since we are looking for \( f^{-1}(0) \), we need to find the value of \( x \) such that \( f(x) = 0 \).

\[ 3x^2 + 24x - 53 = 0 \]

### Step 2: Apply the quadratic formula
The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our case, \( a = 3 \), \( b = 24 \), and \( c = -53 \).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 = 24^2 = 576 \]
\[ 4ac = 4 \times 3 \times (-53) = -636 \]
\[ b^2 - 4ac = 576 + 636 = 1212 \]

### Step 4: Substitute values into the quadratic formula
Now, substituting the values of \( a \), \( b \), and the discriminant into the quadratic formula:

\[ x = \frac{-24 \pm \sqrt{1212}}{2 \times 3} \]

### Step 5: Simplify the expression
Calculating \( \sqrt{1212} \):

\[ \sqrt{1212} \approx 34.8 \quad (\text{exact value is } 2\sqrt{303}) \]

Now substituting back:

\[ x = \frac{-24 \pm 34.8}{6} \]

### Step 6: Calculate the two possible values of \( x \)
Calculating the two values:

1. \( x_1 = \frac{-24 + 34.8}{6} = \frac{10.8}{6} \approx 1.8 \)
2. \( x_2 = \frac{-24 - 34.8}{6} = \frac{-58.8}{6} \approx -9.8 \)

### Step 7: Identify the negative value
The negative value of \( f^{-1}(0) \) is:

\[ x_2 \approx -9.8 \]

Thus, the negative value of \( f^{-1}(0) \) is \( -9.8 \).

### Summary of the solution:
The negative value of \( f^{-1}(0) \) is \( -9.8 \).

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