Find the roots of `3x^(2) - 5x - 12 = 0`.

Answer

To find the roots of the quadratic equation \(3x^2 - 5x - 12 = 0\), we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here, we identify:
- \(a = 3\)
- \(b = -5\)
- \(c = -12\)

### Step 2: Apply the quadratic formula
The quadratic formula to find the roots is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Substitute the values into the formula
Substituting the values of \(a\), \(b\), and \(c\) into the formula:
\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 3 \cdot (-12)}}{2 \cdot 3} \]

### Step 4: Simplify the expression
Calculating the components:
- The term \(-(-5)\) becomes \(5\).
- The term \((-5)^2\) is \(25\).
- The term \(4 \cdot 3 \cdot (-12)\) is \(-144\), so \(b^2 - 4ac\) becomes:
\[ 25 - (-144) = 25 + 144 = 169 \]

Now substituting back into the equation:
\[ x = \frac{5 \pm \sqrt{169}}{6} \]

### Step 5: Calculate the square root
Since \(169\) is a perfect square, we have:
\[ \sqrt{169} = 13 \]

Thus, the equation now becomes:
\[ x = \frac{5 \pm 13}{6} \]

### Step 6: Calculate the two possible values for \(x\)
Now we can find the two roots:
1. For \(x = \frac{5 + 13}{6}\):
\[ x = \frac{18}{6} = 3 \]

2. For \(x = \frac{5 - 13}{6}\):
\[ x = \frac{-8}{6} = -\frac{4}{3} \]

### Conclusion
The roots of the equation \(3x^2 - 5x - 12 = 0\) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{4}{3} \]

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