To solve the quadratic equation [tex]\(3x^2 + 7x - 2 = 0\)[/tex] and describe its roots, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 3\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = -2\)[/tex].
Step 1: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values from our equation:
[tex]\[ \Delta = 7^2 - 4 \cdot 3 \cdot (-2) \][/tex]
[tex]\[ \Delta = 49 + 24 \][/tex]
[tex]\[ \Delta = 73 \][/tex]
Step 2: Evaluate the nature of the discriminant
Since the discriminant [tex]\( \Delta = 73 \)[/tex] is a positive number, it means our quadratic equation has two distinct real roots.
Step 3: Calculate the roots
Using the quadratic formula, we find the roots as follows:
[tex]\[ x = \frac{-7 \pm \sqrt{73}}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-7 \pm \sqrt{73}}{6} \][/tex]
We represent this as:
[tex]\[ x_1 = \frac{-7 + \sqrt{73}}{6} \][/tex]
[tex]\[ x_2 = \frac{-7 - \sqrt{73}}{6} \][/tex]
Step 4: Compare with the given options
The provided options are:
1. [tex]\( -\frac{7}{6} \pm \frac{\sqrt{73}}{6} \)[/tex]
2. [tex]\( -\frac{7}{6} \pm \frac{i \sqrt{73}}{6} \)[/tex]
3. [tex]\( \frac{7}{6} \pm \frac{i \sqrt{73}}{6} \)[/tex]
4. [tex]\( -\frac{7}{6} \pm \frac{\sqrt{73}}{6} \)[/tex]
Here, the correct description matches option 1 and 4:
[tex]\[ -\frac{7}{6} \pm \frac{\sqrt{73}}{6} \][/tex]
Both options 1 and 4 represent the roots correctly as real numbers.
Thus, the correct descriptions are the first and the fourth options:
[tex]\[ -\frac{7}{6} \pm \frac{\sqrt{73}}{6} \][/tex]
This matches the calculated roots perfectly for the equation [tex]\(3x^2 + 7x - 2 = 0\)[/tex].
