Find the values of `x` for which the functions `f(x)=3x^2-1` and `g(x)=3+x` are equal

Options

A. `[-1,(4)/(3)]`
B. `[1,(4)/(3)]`
C. `[-1,-(4)/(3)]`
D. `[-2,-(4)/(3)]`

Explanation

To find the values of \( x \) for which the functions \( f(x) = 3x^2 - 1 \) and \( g(x) = 3 + x \) are equal, we will follow these steps:

### Step 1: Set the functions equal to each other
We start by setting the two functions equal:
\[ f(x) = g(x) \]
This gives us the equation:
\[ 3x^2 - 1 = 3 + x \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ 3x^2 - x - 4 = 0 \]

### Step 3: Identify the coefficients
Now we identify the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \):
- \( a = 3 \)
- \( b = -1 \)
- \( c = -4 \)

### Step 4: Factor the quadratic equation
We need to factor the quadratic equation \( 3x^2 - x - 4 = 0 \). We look for two numbers that multiply to \( ac = 3 \times -4 = -12 \) and add up to \( b = -1 \). The numbers that satisfy this are \( -4 \) and \( 3 \).

We can rewrite the middle term:
\[ 3x^2 - 4x + 3x - 4 = 0 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (3x^2 - 4x) + (3x - 4) = 0 \]

### Step 6: Factor by grouping
Now we factor by grouping:
\[ x(3x - 4) + 1(3x - 4) = 0 \]
This can be factored as:
\[ (3x - 4)(x + 1) = 0 \]

### Step 7: Solve for \( x \)
Now we set each factor equal to zero:
1. \( 3x - 4 = 0 \)
\[ 3x = 4 \implies x = \frac{4}{3} \]
2. \( x + 1 = 0 \)
\[ x = -1 \]

### Final Answer
Thus, the values of \( x \) for which \( f(x) \) and \( g(x) \) are equal are:
\[ x = -1 \quad \text{and} \quad x = \frac{4}{3} \]

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