The Solution To 3x2 12x 24 0 Is

The Solution to 3x² + 12x + 24 = 0: A Comprehensive Exploration

Solving quadratic equations is a fundamental concept in algebra, forming the bedrock for many advanced mathematical concepts. This article delves deep into finding the solution to the quadratic equation 3x² + 12x + 24 = 0, exploring various methods and providing a thorough understanding of the process. We'll not only find the solution but also unpack the underlying theory and demonstrate how to approach similar problems.

Understanding Quadratic Equations

Before jumping into the solution, let's refresh our understanding of quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to this equation, also known as the roots or zeros, represent the x-values where the quadratic function intersects the x-axis. These roots can be real numbers, complex numbers, or a combination of both.

Our specific equation, 3x² + 12x + 24 = 0, fits this standard form with a = 3, b = 12, and c = 24.

Method 1: Factoring

Factoring is a method used to simplify the equation by expressing it as a product of simpler expressions. This method is particularly efficient when the quadratic expression can be easily factored. However, not all quadratic equations are easily factorable. Let's try factoring our equation:

First, we can simplify the equation by dividing the entire equation by 3:

x² + 4x + 8 = 0

Now, let's attempt to find two numbers that add up to 4 (the coefficient of x) and multiply to 8 (the constant term). Unfortunately, no such integers exist. This means that the quadratic equation cannot be easily factored using integers. Therefore, we need to explore alternative methods.

Method 2: The Quadratic Formula

The quadratic formula provides a direct and universally applicable method for solving quadratic equations, regardless of their factorability. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this formula to our equation (using the simplified version x² + 4x + 8 = 0, where a = 1, b = 4, and c = 8):

x = [-4 ± √(4² - 4 * 1 * 8)] / (2 * 1)

x = [-4 ± √(16 - 32)] / 2

x = [-4 ± √(-16)] / 2

Notice that we have a negative number under the square root. This indicates that the solutions to the equation are complex numbers.

Understanding Complex Numbers

Complex numbers involve the imaginary unit 'i', defined as the square root of -1 (i = √-1). We can rewrite the square root of -16 as follows:

√(-16) = √(16 * -1) = √16 * √-1 = 4i

Substituting this back into the quadratic formula:

x = [-4 ± 4i] / 2

This simplifies to:

x = -2 ± 2i

Therefore, the solutions to the equation 3x² + 12x + 24 = 0 are x = -2 + 2i and x = -2 - 2i. These are two distinct complex conjugate roots.

Method 3: Completing the Square

Completing the square is another algebraic technique to solve quadratic equations. This method involves manipulating the equation to form a perfect square trinomial, which can then be easily factored.

Starting with the simplified equation x² + 4x + 8 = 0:

1. Move the constant term to the right side: x² + 4x = -8

2. Take half of the coefficient of x (which is 4), square it (2² = 4), and add it to both sides: x² + 4x + 4 = -8 + 4 x² + 4x + 4 = -4

3. Factor the left side as a perfect square: (x + 2)² = -4

4. Take the square root of both sides: x + 2 = ±√(-4) x + 2 = ±2i

5. Solve for x: x = -2 ± 2i

Again, we arrive at the same solutions: x = -2 + 2i and x = -2 - 2i.

Graphical Representation

While the solutions are complex numbers, it's insightful to visualize the quadratic function graphically. The parabola represented by y = 3x² + 12x + 24 will not intersect the x-axis, confirming that there are no real roots. The complex roots represent the points where the parabola would intersect the x-axis if we extended the graph into the complex plane.

Verifying the Solutions

It's crucial to verify our solutions by substituting them back into the original equation:

For x = -2 + 2i:

3(-2 + 2i)² + 12(-2 + 2i) + 24 = 3(4 - 8i - 4) + (-24 + 24i) + 24 = -24i + 24i = 0

For x = -2 - 2i:

3(-2 - 2i)² + 12(-2 - 2i) + 24 = 3(4 + 8i - 4) + (-24 - 24i) + 24 = 24i - 24i = 0

Both solutions satisfy the original equation, confirming their accuracy.

Applications of Quadratic Equations with Complex Roots

Although it might seem that complex roots are less significant than real roots, they have crucial applications in various fields, including:

• Electrical Engineering: Complex numbers are essential in analyzing alternating current (AC) circuits, where impedance and phase angles are involved.
• Quantum Mechanics: Complex numbers are fundamental in describing wave functions and probabilities in quantum systems.
• Signal Processing: Complex numbers facilitate the analysis and manipulation of signals in various applications.
• Fluid Dynamics: Complex analysis is used to solve certain fluid flow problems.

Conclusion

Solving the quadratic equation 3x² + 12x + 24 = 0 demonstrates the importance of understanding different solution methods and interpreting results. While the equation does not have real roots, the complex roots obtained through the quadratic formula and completing the square are equally valid and have significant implications in diverse fields of study. This comprehensive exploration underscores the power and versatility of algebraic techniques in solving complex mathematical problems. Remember to always verify your solutions and understand the broader context of the results you obtain. Mastering quadratic equations is a crucial step in your mathematical journey, opening doors to more advanced concepts and applications.