For What Values Of X Is X2 2x 24 True

For What Values of x is x² + 2x - 24 = 0 True? A Comprehensive Exploration

Finding the values of x that satisfy the quadratic equation x² + 2x - 24 = 0 is a fundamental problem in algebra. This seemingly simple equation unlocks a deeper understanding of quadratic functions, their graphs, and their applications in various fields. This article will explore multiple methods to solve this equation, delve into the underlying concepts, and discuss the significance of the solutions found.

Understanding Quadratic Equations

Before diving into the solution, let's establish a foundational 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 highest power of the variable x is 2, hence the term "quadratic." The solutions to a quadratic equation are also known as its roots, zeros, or x-intercepts. These values represent the points where the graph of the quadratic function intersects the x-axis.

Our specific equation, x² + 2x - 24 = 0, fits this general form with a = 1, b = 2, and c = -24.

Method 1: Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation is easily factorable. The goal is to rewrite the quadratic expression as a product of two linear expressions.

We look for two numbers that add up to 'b' (2) and multiply to 'c' (-24). These numbers are 6 and -4. Therefore, we can factor the equation as follows:

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

This equation is true if either (x + 6) = 0 or (x - 4) = 0. Solving these linear equations gives us the solutions:

• x = -6
• x = 4

These are the two values of x that satisfy the original quadratic equation.

Method 2: Quadratic Formula

The quadratic formula is a more general method that works for all quadratic equations, even those that are not easily factorable. The formula is derived from completing the square and provides a direct way to calculate the roots:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 2, c = -24) into the quadratic formula:

x = [-2 ± √(2² - 4 * 1 * -24)] / (2 * 1) x = [-2 ± √(4 + 96)] / 2 x = [-2 ± √100] / 2 x = [-2 ± 10] / 2

This gives us two solutions:

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

As expected, we obtain the same solutions as with the factoring method.

Method 3: Completing the Square

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

Starting with x² + 2x - 24 = 0:

1. Move the constant term to the right side: x² + 2x = 24
2. Take half of the coefficient of x (which is 2), square it (1), and add it to both sides: x² + 2x + 1 = 24 + 1
3. Factor the left side as a perfect square: (x + 1)² = 25
4. Take the square root of both sides: x + 1 = ±5
5. Solve for x:
   • x + 1 = 5 => x = 4
   • x + 1 = -5 => x = -6

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

Graphical Representation

The solutions to the quadratic equation x² + 2x - 24 = 0 correspond to the x-intercepts of the parabola represented by the function y = x² + 2x - 24. Graphing this function reveals that the parabola intersects the x-axis at x = -6 and x = 4, visually confirming our solutions.

Significance of the Solutions

The solutions x = 4 and x = -6 represent the points where the quadratic function equals zero. These values are crucial in various applications:

• Finding roots of a function: In mathematics, finding the roots of a function is fundamental to understanding its behavior and properties.
• Solving real-world problems: Quadratic equations model many real-world phenomena, such as projectile motion, area calculations, and optimization problems. The solutions provide crucial insights into these situations. For example, if the equation represents the height of a projectile over time, the roots would indicate the times when the projectile is at ground level.
• Engineering and Physics: Quadratic equations are extensively used in engineering and physics for modeling various systems and analyzing their behavior. The solutions help determine critical parameters and design specifications.

Analyzing the Discriminant

The discriminant (b² - 4ac) in the quadratic formula provides valuable information about the nature of the solutions:

• Discriminant > 0: Two distinct real roots (as in our case).
• Discriminant = 0: One real root (a repeated root).
• Discriminant < 0: Two complex conjugate roots (no real solutions).

In our equation, the discriminant is 100 (which is > 0), indicating two distinct real roots, which we have already found to be 4 and -6.

Conclusion

The quadratic equation x² + 2x - 24 = 0 has two solutions: x = 4 and x = -6. We have explored three different methods—factoring, the quadratic formula, and completing the square—to arrive at these solutions. Understanding these methods and the underlying concepts is essential for solving more complex quadratic equations and for applying them to various mathematical and real-world problems. The graphical representation and analysis of the discriminant further enhance our comprehension of the nature and significance of these solutions. This comprehensive exploration demonstrates the versatility and importance of quadratic equations in diverse fields of study.