Find nature of roots and roots of x²-3√3-12

The given quadratic equation is:

\[
x^2 - 3\sqrt{3}x - 12 = 0
\]

### 1. Nature of Roots:
To determine the nature of the roots, we use the **discriminant formula**:

\[
\Delta = b^2 - 4ac
\]

For the equation \(ax^2 + bx + c = 0\), we have:
- \(a = 1\)
- \(b = -3\sqrt{3}\)
- \(c = -12\)

Substitute these values into the discriminant formula:

\[
\Delta = (-3\sqrt{3})^2 - 4(1)(-12)
\]
\[
\Delta = 27 + 48 = 75
\]

Since \(\Delta > 0\), the roots are **real and distinct**.

### 2. Finding the Roots:
To find the roots, use the **quadratic formula**:

\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\]

Substitute the values of \(a\), \(b\), and \(\Delta\):

\[
x = \frac{-(-3\sqrt{3}) \pm \sqrt{75}}{2(1)}
\]
\[
x = \frac{3\sqrt{3} \pm \sqrt{75}}{2}
\]

Now, simplify \(\sqrt{75}\):
\[
\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}
\]

So the roots are:
\[
x = \frac{3\sqrt{3} \pm 5\sqrt{3}}{2}
\]

Thus, the two roots are:

1. \(x_1 = \frac{3\sqrt{3} + 5\sqrt{3}}{2} = \frac{8\sqrt{3}}{2} = 4\sqrt{3}\)

2. \(x_2 = \frac{3\sqrt{3} - 5\sqrt{3}}{2} = \frac{-2\sqrt{3}}{2} = -\sqrt{3}\)

### Final Answer:
- **Nature of roots**: Real and distinct.
- **Roots**: \(x_1 = 4\sqrt{3}\) and \(x_2 = -\sqrt{3}\).

Do you need more details or have any questions?

### Related Questions:
1. How do you determine the discriminant of a quadratic equation?
2. What happens if the discriminant is negative?
3. How is the quadratic formula derived?
4. Can the quadratic formula solve non-real roots?
5. What are the applications of quadratic equations in real-life problems?

### Tip:
Always check the discriminant first to understand the nature of the roots before solving the equation.

Find nature of roots and roots of x² - 3√3 - 12

Discover how to find the nature of the roots and solve the quadratic equation x² - 3√3x - 12. This problem demonstrates key algebraic concepts such as the discriminant, quadratic formula, and finding real and distinct roots. Perfect for students in Grades 9-10.

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