The zeroes of `3x^(2)-4-x` are :

Options

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

Explanation

To find the zeroes of the polynomial \(3x^2 - 4 - x\), we will follow these steps:

### Step 1: Set the polynomial equal to zero
We start by setting the polynomial equal to zero to find its roots:
\[ 3x^2 - x - 4 = 0 \]

### Step 2: Rearrange the equation
We can rearrange the equation into standard form:
\[ 3x^2 - x - 4 = 0 \]

### Step 3: Use the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 3\), \(b = -1\), and \(c = -4\).

### Step 4: Calculate the discriminant
First, we calculate the discriminant \(D\):
\[ D = b^2 - 4ac = (-1)^2 - 4 \cdot 3 \cdot (-4) = 1 + 48 = 49 \]

### Step 5: Substitute into the quadratic formula
Now we substitute \(a\), \(b\), and \(D\) into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{49}}{2 \cdot 3} = \frac{1 \pm 7}{6} \]

### Step 6: Calculate the two possible values for \(x\)
Now we calculate the two possible values for \(x\):
1. \(x = \frac{1 + 7}{6} = \frac{8}{6} = \frac{4}{3}\)
2. \(x = \frac{1 - 7}{6} = \frac{-6}{6} = -1\)

### Step 7: State the zeroes
Thus, the zeroes of the polynomial \(3x^2 - x - 4\) are:
\[ x = -1 \quad \text{and} \quad x = \frac{4}{3} \]

### Final Answer
Therefore, the zeroes of the polynomial are \(x = -1\) and \(x = \frac{4}{3}\).

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