The Solution To X2 10x 24 Is

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

Solving quadratic equations is a fundamental concept in algebra, and understanding the different methods is crucial for success in mathematics. This article will delve into the solution of the quadratic equation x² + 10x + 24 = 0, exploring various approaches and providing a detailed explanation of each step. We'll go beyond simply finding the answer, focusing on the underlying principles and techniques applicable to a wider range of quadratic equations.

Understanding 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. Our specific equation, x² + 10x + 24 = 0, fits this form with a = 1, b = 10, and c = 24. The solutions to this equation, also known as the roots or zeros, represent the x-values where the corresponding quadratic function intersects the x-axis.

Method 1: Factoring

Factoring is a straightforward method for solving quadratic equations, particularly when the equation is easily factorable. This method relies on finding two numbers that add up to 'b' (the coefficient of x) and multiply to 'c' (the constant term).

In our equation, x² + 10x + 24 = 0, we need to find two numbers that add up to 10 and multiply to 24. These numbers are 6 and 4. Therefore, we can factor the equation as follows:

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

This factored form tells us that the product of two terms, (x + 6) and (x + 4), is equal to zero. For this to be true, at least one of the terms must be equal to zero. This leads to two possible solutions:

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

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

Advantages of Factoring: It's a quick and efficient method when applicable. It provides a clear understanding of the equation's structure and roots.

Disadvantages of Factoring: Not all quadratic equations are easily factorable, especially those with irrational or complex roots.

Method 2: Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations, applicable even when factoring is difficult or impossible. The formula is derived from completing the square and provides the solutions for any quadratic equation of the form ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this formula to our equation, x² + 10x + 24 = 0, where a = 1, b = 10, and c = 24:

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

As expected, we obtain the same solutions as with the factoring method: x = -6 and x = -4.

Advantages of the Quadratic Formula: It works for all quadratic equations, regardless of their factorability.

Disadvantages of the Quadratic Formula: It can be more computationally intensive than factoring, especially if the equation involves large numbers or irrational coefficients.

Method 3: Completing the Square

Completing the square is another powerful method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Let's apply this method to x² + 10x + 24 = 0:

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

2. Take half of the coefficient of x (10/2 = 5), square it (5² = 25), and add it to both sides: x² + 10x + 25 = -24 + 25 x² + 10x + 25 = 1

3. Factor the left side as a perfect square trinomial: (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

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

Advantages of Completing the Square: It provides a deeper understanding of the relationship between the equation's coefficients and its roots. It's useful in various areas of mathematics beyond solving quadratic equations, such as deriving the quadratic formula itself and working with conic sections.

Disadvantages of Completing the Square: It can be more tedious than factoring or using the quadratic formula, especially for equations with fractional coefficients.

Graphical Representation

The solutions to the quadratic equation x² + 10x + 24 = 0 represent the x-intercepts of the parabola defined by the function y = x² + 10x + 24. Graphing this function visually confirms our solutions. The parabola will intersect the x-axis at x = -4 and x = -6. This graphical representation provides a visual interpretation of the equation's roots and helps solidify the understanding of the relationship between the algebraic solution and the geometric representation of the quadratic function.

Discriminant and Nature of Roots

The discriminant, represented by Δ (delta) and calculated as b² - 4ac, plays a crucial role in determining the nature of the roots of a quadratic equation.

• Δ > 0: Two distinct real roots.
• Δ = 0: One real root (a repeated root).
• Δ < 0: Two complex conjugate roots.

For our equation, x² + 10x + 24 = 0, the discriminant is:

Δ = 10² - 4 * 1 * 24 = 100 - 96 = 4

Since Δ > 0, we confirm that there are two distinct real roots, which we found to be -4 and -6.

Applications of Quadratic Equations

Quadratic equations have wide-ranging applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains in materials.
• Economics: Modeling supply and demand, optimizing production costs.
• Computer Graphics: Creating curved shapes and animations.

Conclusion

Solving the quadratic equation x² + 10x + 24 = 0 demonstrates the versatility of different algebraic techniques. Whether using factoring, the quadratic formula, or completing the square, understanding these methods provides a strong foundation for tackling more complex mathematical problems. The discriminant helps predict the nature of the roots, providing valuable insight into the solution's characteristics. The graphical representation offers a visual interpretation, connecting the algebraic solutions to the geometric properties of the quadratic function. The widespread applicability of quadratic equations underscores their importance in various scientific and practical fields. Mastering these techniques is essential for success in higher-level mathematics and related disciplines.