The Solution To X2 10x 24 Is

The Solution to x² + 10x + 24 = 0: A Comprehensive Guide

Solving quadratic equations is a fundamental skill in algebra, crucial for various applications in mathematics, science, and engineering. This article delves into the solution of the specific quadratic equation x² + 10x + 24 = 0, exploring multiple methods to arrive at the correct answer and providing a detailed explanation of the underlying principles. We’ll cover factoring, the quadratic formula, completing the square, and graphing, demonstrating their application and highlighting their strengths and weaknesses for different types of quadratic equations.

Understanding Quadratic Equations

Before diving into the solution, let's understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). In our specific case, x² + 10x + 24 = 0, we have a = 1, b = 10, and c = 24.

Method 1: Factoring

Factoring is often the quickest and most efficient method for solving quadratic equations, especially when the equation has easily identifiable factors. The goal is to rewrite the quadratic expression as a product of two binomials.

For x² + 10x + 24 = 0, we look for two numbers that add up to 10 (the coefficient of x) and multiply to 24 (the constant term). These numbers are 6 and 4. Therefore, we can factor the equation as follows:

(x + 6)(x + 4) = 0

This equation is satisfied if either (x + 6) = 0 or (x + 4) = 0. Solving for x in each case gives us:

• x + 6 = 0 => x = -6
• x + 4 = 0 => x = -4

Therefore, the solutions to the quadratic equation x² + 10x + 24 = 0 are x = -6 and x = -4.

When Factoring is Difficult

Factoring isn't always straightforward. Some quadratic equations have irrational or complex roots, making factoring impractical. In such cases, other methods are necessary.

Method 2: The Quadratic Formula

The quadratic formula is a powerful tool that provides a solution for any quadratic equation, regardless of the nature of its roots. It's derived from completing the square (explained in the next section) and is given by:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 10, c = 24) into the quadratic formula, we get:

x = [-10 ± √(10² - 4 * 1 * 24)] / (2 * 1) x = [-10 ± √(100 - 96)] / 2 x = [-10 ± √4] / 2 x = [-10 ± 2] / 2

This gives us two solutions:

• x = (-10 + 2) / 2 = -8 / 2 = -4
• x = (-10 - 2) / 2 = -12 / 2 = -6

Again, we arrive at the solutions x = -6 and x = -4.

Method 3: Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This method is particularly useful when factoring is difficult or impossible.

The process involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

1. Move the constant term to the right side: x² + 10x = -24

2. Take half of the coefficient of x (which is 10), square it (25), and add it to both sides: x² + 10x + 25 = -24 + 25 x² + 10x + 25 = 1

3. Factor the perfect square trinomial on the left side: (x + 5)² = 1

4. Take the square root of both sides: x + 5 = ±√1 x + 5 = ±1

5. Solve for x: x = -5 + 1 = -4 x = -5 - 1 = -6

Once again, we obtain the solutions x = -4 and x = -6.

Method 4: Graphing

Graphing a quadratic equation provides a visual representation of its solutions. The solutions are the x-intercepts (where the graph crosses the x-axis). While not as precise as algebraic methods for finding exact solutions, graphing is excellent for visualizing the behavior of the quadratic function and estimating the solutions.

To graph y = x² + 10x + 24, you can plot points or use a graphing calculator or software. The graph will be a parabola. The x-intercepts, where y = 0, represent the solutions to the equation x² + 10x + 24 = 0. You'll find the parabola intersects the x-axis at x = -4 and x = -6.

The Discriminant and the Nature of Roots

The expression b² - 4ac within the quadratic formula is called the discriminant. It provides information about the nature of the roots:

• b² - 4ac > 0: The equation has two distinct real roots.
• b² - 4ac = 0: The equation has one real root (a repeated root).
• b² - 4ac < 0: The equation has two complex roots (roots involving the imaginary unit 'i').

In our case, b² - 4ac = 10² - 4(1)(24) = 4, which is greater than 0. This confirms that the equation x² + 10x + 24 = 0 has two distinct real roots, as we've found.

Applications of Quadratic Equations

Understanding how to solve quadratic equations is vital in various fields:

• Physics: Calculating projectile motion, analyzing oscillations, and solving problems related to gravity.
• Engineering: Designing structures, analyzing circuits, and optimizing systems.
• Economics: Modeling supply and demand, optimizing production, and analyzing market trends.
• Computer Science: Developing algorithms, creating graphics, and solving optimization problems.

Conclusion

Solving the quadratic equation x² + 10x + 24 = 0 demonstrates the power and versatility of different algebraic techniques. Factoring provides a quick solution when possible, while the quadratic formula offers a universal approach. Completing the square provides a deeper understanding of the underlying mathematical principles, and graphing offers a visual representation of the solutions. Understanding the discriminant allows you to predict the nature of the roots before even solving the equation. Mastering these methods is essential for success in mathematics and its numerous applications in diverse fields. Remember to choose the most appropriate method based on the characteristics of the equation and the desired level of precision.