Finding the Nature of Roots for $3x^2 + 2x + 5 = 0$

To determine the nature of the roots of a quadratic equation, we examine its discriminant. The given quadratic equation is:

$$3x^2 + 2x + 5 = 0$$

Understanding the Discriminant

A quadratic equation is generally represented in the form:

$$ax^2 + bx + c = 0$$

Where $a$, $b$, and $c$ are coefficients. The discriminant, denoted by $\Delta$, is calculated using the formula:

$$\Delta = b^2 - 4ac$$

The value of the discriminant helps us classify the roots:

• If $\Delta > 0$, the roots are real and distinct.
• If $\Delta = 0$, the roots are real and equal.
• If $\Delta < 0$, the roots are complex (or imaginary) and not real.

Calculating the Discriminant

For the given equation $3x^2 + 2x + 5 = 0$, we identify the coefficients:

• $a = 3$
• $b = 2$
• $c = 5$

Now, let's substitute these values into the discriminant formula:

$$\Delta = (2)^2 - 4(3)(5)$$

First, calculate $b^2$:

$$\Delta = 4 - 4(3)(5)$$

Next, calculate $4ac$:

$$4(3)(5) = 12 \times 5 = 60$$

Now, subtract $4ac$ from $b^2$:

$$\Delta = 4 - 60$$

$$\Delta = -56$$

Interpreting the Result

The calculated discriminant is $\Delta = -56$. Since $-56$ is less than $0$ ($\Delta < 0$), the roots of the quadratic equation $3x^2 + 2x + 5 = 0$ are complex (imaginary).

Conclusion

Comparing our result with the given options, the nature of the roots is Complex Roots (Imaginary Roots).