Solving the Quadratic Equation 2x2 - 2√3x 3 0: A Comprehensive Guide

In this article, we will explore how to solve the quadratic equation 2x2 - 2√3x 3 0 using the quadratic formula and gain insight into the roots of the equation. Understanding how to solve such equations is crucial in various fields, including mathematics, physics, and engineering. Let's delve into the process step by step.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically represented as ax2 bx c 0. In our case, the equation is 2x2 - 2√3x 3 0. Here, the coefficients are a 2, b -2√3, and c -3.

Using the Quadratic Formula

To find the roots of the quadratic equation, we can use the quadratic formula:

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

Let's plug in the values of a, b, and c into the formula:

a 2, b -2√3, and c -3

Step 1: Calculate the Discriminant

The discriminant, denoted as D, is given by:

[D b^2 - 4ac]

Substituting the values:

[D (-2√3)^2 - 4(2)(-3)]

[D 4(3) 24]

[D 12 24 36 - 12 -12]

Since the discriminant D is negative, the roots of the equation are complex numbers.

Step 2: Substitute into the Quadratic Formula

Now, let's substitute b, a, and c into the quadratic formula:

[x frac{-(-2√3) pm sqrt{-12}}{2(2)}]

[x frac{2√3 pm sqrt{-12}}{4}]

Note that sqrt{-12} can be rewritten as 2√3i because sqrt{-12} sqrt{-1 * 12} sqrt{-1} * sqrt{12} i * 2√3 2√3i.

Thus, the roots are:

[x frac{2√3 pm 2√3i}{4} frac{√3 pm √3i}{2}]

Therefore, the roots of the equation 2x2 - 2√3x 3 0 are:

[x frac{√3}{2} frac{√3}{2}i]

[x frac{√3}{2} - frac{√3}{2}i]

Applications and Significance

The roots of such quadratic equations are particularly important in various scientific and engineering applications. They help in understanding the behavior of physical systems, analyzing electrical circuits, and solving optimization problems. Knowledge of complex roots also plays a vital role in advanced mathematics, such as in the study of complex analysis and number theory.

Conclusion

By following the steps outlined above, we can solve the quadratic equation 2x2 - 2√3x 3 0 and find its roots. The use of the quadratic formula and the discriminant is a powerful tool in solving such equations, even when the roots are complex. Understanding these concepts is essential for anyone interested in mathematical problem-solving and application.