The roots of the equation `12 x^2 + x - 1 = 0` is :

Explanation

To find the roots of the quadratic equation \(12x^2 + x - 1 = 0\), we can use the quadratic formula, which is given by:

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

where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\).

### Step 1: Identify the coefficients
From the equation \(12x^2 + x - 1 = 0\), we can identify:
- \(a = 12\)
- \(b = 1\)
- \(c = -1\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]

Substituting the values of \(a\), \(b\), and \(c\):
\[ D = 1^2 - 4 \cdot 12 \cdot (-1) = 1 + 48 = 49 \]

### Step 3: Apply the quadratic formula
Now, we can substitute the values of \(a\), \(b\), and \(D\) into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{49}}{2 \cdot 12} \]

Calculating \(\sqrt{49}\):
\[ \sqrt{49} = 7 \]

Now substituting this back into the formula:
\[ x = \frac{-1 \pm 7}{24} \]

### Step 4: Calculate the two roots
Now we will calculate the two possible values for \(x\):
1. For the positive case:
\[ x_1 = \frac{-1 + 7}{24} = \frac{6}{24} = \frac{1}{4} \]

2. For the negative case:
\[ x_2 = \frac{-1 - 7}{24} = \frac{-8}{24} = \frac{-1}{3} \]

### Conclusion
The roots of the equation \(12x^2 + x - 1 = 0\) are:
\[ x = \frac{1}{4} \quad \text{and} \quad x = -\frac{1}{3} \]

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