Solving the Quadratic Equation 3x2 2x 4 0 Using the Quadratic Formula

In this article, we will discuss in detail how to solve the quadratic equation 3x2 2x 4 0 using the quadratic formula. This is a valuable skill in mathematics, and understanding it will provide insight into solving other complex equations.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax2 bx c 0, where a, b, and c are constants, and a ≠ 0. The formula is given by

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

Applying the Quadratic Formula to 3x2 2x 4 0

Given the equation 3x2 2x 4 0, we need to identify the coefficients a, b, and c. Here, a 3, b 2, and c 4.

Substituting these values into the quadratic formula, we get:

[ x frac{-2 pm sqrt{2^2 - 4 cdot 3 cdot 4}}{2 cdot 3} ]

First, let's simplify the expression under the square root:

[ b^2 - 4ac 2^2 - 4 cdot 3 cdot 4 4 - 48 -44 ]

The discriminant is -44, which is negative. A negative discriminant indicates that the solutions to the equation are complex numbers.

Now, let's complete the calculation:

[ x frac{-2 pm sqrt{-44}}{6} ]

We can further simplify the square root of a negative number by expressing it in terms of the imaginary unit i, where i2 -1.

[ sqrt{-44} sqrt{44}i 2sqrt{11}i ]

Thus, the solutions are:

[ x frac{-2 pm 2sqrt{11}i}{6} ]

This can be further simplified to:

[ x frac{-1 pm sqrt{11}i}{3} ]

Conclusion

The solutions to the equation 3x2 2x 4 0 are given by:

[ x frac{-1 sqrt{11}i}{3} text{ and } x frac{-1 - sqrt{11}i}{3} ]

Understanding the quadratic formula and its application to complex equations is crucial, especially in higher mathematics and engineering applications. If you found this explanation helpful and would like to see more such explanations, please give a thumbs up and follow for more updates.