Solving the Quadratic Equation 3x72^24x-3: A Comprehensive Guide

Quadratic equations form the backbone of algebra and are frequently encountered in various mathematical and real-world applications. One such equation is 3x72^24x-3. This article provides a step-by-step guide to solving this particular quadratic equation, explaining each step to ensure clarity and understanding.

Understanding the Quadratic Equation

The given equation is 3x72^24x-3. To solve this, we first need to rewrite it in a standard form. The equation can be rearranged as follows:

3x72^2 4x - 3

This equation can be further modified into a standard quadratic form:

9x^2 - 432x 5187 0

Step-by-Step Solution

To solve the equation 9x^2 - 432x 5187 0, we can use the quadratic formula, which is given by:

x [-b ± sqrt(b^2 - 4ac)] / 2a

In our equation, a 9, b -432, and c 5187. Plugging these values into the quadratic formula, we get:

x [432 ± sqrt((-432)^2 - 4*9*5187)] / (2*9)

Let's break this down step by step:

Step 1: Calculate the discriminant

Δ b^2 - 4ac

Substituting the values of a, b, and c:

Δ (-432)^2 - 4*9*5187

Δ 186624 - 186732

Δ -108

Step 2: Use the quadratic formula

x [432 ± sqrt(-108)] / 18

Since the discriminant is negative, the solutions will involve complex numbers:

x1 [432 sqrt(-108)] / 18

x2 [432 - sqrt(-108)] / 18

Step 3: Simplify the solutions

Let's simplify x1 and x2:

x1 [432 sqrt(108) * i] / 18

x2 [432 - sqrt(108) * i] / 18

Where i is the imaginary unit, defined by i^2 -1. Further simplifying the square root:

x1 [432 6sqrt(3) * i] / 18

x2 [432 - 6sqrt(3) * i] / 18

Finally, simplifying the fractions:

x1 (24 sqrt(3) * i)

x2 (24 - sqrt(3) * i)

Conclusion

The solutions to the equation 3x72^24x-3 are complex and involve the imaginary unit i. The real solutions do not exist in this case, highlighting the importance of understanding complex numbers in solving quadratic equations.

F Unexpected Applications

Quadratic equations are not just theoretical constructs but have real-world applications. For instance, they are used in physics to describe parabolic motion, in economics to model supply and demand scenarios, and in engineering to design structures that withstand various forces.

References

For further reading on quadratic equations and their applications, refer to the following resources:

Linear Algebra and Its Applications by Gilbert Strang Understanding Algebra by James Sellers Algebra: Structure and Method by Richard G. Brown

QA

Q: Can a quadratic equation always be solved using the quadratic formula?

A: Yes, the quadratic formula can be applied to any quadratic equation of the form ax^2 bx c 0, providing the discriminant is not negative.

Q: What is the geometric interpretation of the roots of a quadratic equation?

A: The roots of a quadratic equation represent the points where the corresponding parabola crosses the x-axis.

Q: How do complex roots occur, and what does it mean in real-world applications?

A: Complex roots occur when the discriminant is negative, indicating that the parabola does not cross the x-axis in the real plane. In real-world applications, complex roots often signify conditions that are idealized or theoretical rather than physically observable.