Solving For X In X^2 + 2x = 24
Introduction
In this comprehensive article, we will delve into the process of solving the quadratic equation x^2 + 2x = 24. Quadratic equations are fundamental in algebra and have wide-ranging applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone studying mathematics or related disciplines. This article aims to provide a step-by-step guide to finding the values of x that satisfy the given equation, ensuring clarity and a thorough understanding of the methods involved. By the end of this discussion, you will not only know the correct solutions but also grasp the underlying principles and techniques applicable to solving other quadratic equations.
Understanding Quadratic Equations
Before diving into the solution, it's essential to understand what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are the values of x that make the equation true. These solutions are also known as roots or zeros of the quadratic equation. Quadratic equations can have two, one, or no real solutions, depending on the discriminant (b^2 - 4ac). The discriminant provides insight into the nature of the roots, helping us understand whether they are real and distinct, real and equal, or complex.
Transforming the Equation
The given equation is x^2 + 2x = 24. To solve it, we first need to transform it into the standard form ax^2 + bx + c = 0. This is achieved by subtracting 24 from both sides of the equation. This step is crucial because it sets the stage for applying various solution methods such as factoring, completing the square, or using the quadratic formula. Transforming the equation ensures that all terms are on one side, making it easier to identify the coefficients a, b, and c, which are essential for further analysis and application of solution techniques. The transformed equation is a prerequisite for any standard method of solving quadratic equations, ensuring a systematic and accurate approach.
Subtracting 24 from both sides, we get:
x^2 + 2x - 24 = 0
Now, the equation is in the standard form, where a = 1, b = 2, and c = -24. This transformation allows us to apply various methods for solving quadratic equations effectively.
Method 1: Factoring
One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. This method is particularly effective when the quadratic expression can be easily factored. The goal is to find two numbers that multiply to c (in this case, -24) and add up to b (in this case, 2). These numbers will be used to construct the binomial factors. Factoring simplifies the equation into a form where we can easily identify the solutions by setting each factor equal to zero. This method not only provides a straightforward way to find the roots but also enhances understanding of the relationship between the coefficients and the roots of the quadratic equation.
Finding the Factors
We need to find two numbers that multiply to -24 and add to 2. Let's list the factor pairs of -24:
- (-1, 24)
- (1, -24)
- (-2, 12)
- (2, -12)
- (-3, 8)
- (3, -8)
- (-4, 6)
- (4, -6)
From the list, we can see that the pair -4 and 6 satisfy both conditions: (-4) * 6 = -24 and (-4) + 6 = 2. These numbers are the key to factoring the quadratic expression.
Factoring the Quadratic Expression
Now that we have the numbers -4 and 6, we can rewrite the quadratic expression as:
x^2 + 2x - 24 = (x - 4)(x + 6)
This step transforms the quadratic equation into a product of two binomials, making it easier to find the solutions.
Solving for x
To find the values of x that make the equation true, we set each factor equal to zero:
- x - 4 = 0 Adding 4 to both sides, we get: x = 4
- x + 6 = 0 Subtracting 6 from both sides, we get: x = -6
Therefore, the solutions to the quadratic equation x^2 + 2x = 24 are x = 4 and x = -6.
Method 2: Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax^2 + bx + c = 0. This formula provides the solutions for x regardless of whether the equation can be easily factored. The quadratic formula is particularly useful when the roots are not rational or when factoring is difficult. It is derived from the method of completing the square and is a fundamental tool in algebra. The formula guarantees a solution for any quadratic equation, making it a reliable method in various mathematical and scientific applications. Understanding and applying the quadratic formula is essential for anyone studying algebra and related fields.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation.
Applying the Quadratic Formula
In our equation, x^2 + 2x - 24 = 0, we have a = 1, b = 2, and c = -24. Plugging these values into the quadratic formula, we get:
x = (-2 ± √(2^2 - 4 * 1 * (-24))) / (2 * 1)
Simplifying the Expression
Let's simplify the expression step by step:
x = (-2 ± √(4 + 96)) / 2
x = (-2 ± √100) / 2
x = (-2 ± 10) / 2
Finding the Solutions
Now, we have two possible solutions:
- x = (-2 + 10) / 2 x = 8 / 2 x = 4
- x = (-2 - 10) / 2 x = -12 / 2 x = -6
Thus, using the quadratic formula, we find the solutions to be x = 4 and x = -6, which confirms our previous result from factoring.
Conclusion
In this article, we successfully found the values of x that satisfy the quadratic equation x^2 + 2x = 24. We explored two methods: factoring and the quadratic formula. Both methods led us to the same solutions: x = 4 and x = -6. Factoring provided a straightforward approach by expressing the quadratic expression as a product of binomials, while the quadratic formula offered a universal method applicable to any quadratic equation. Understanding these methods not only helps in solving quadratic equations but also enhances problem-solving skills in algebra and related fields. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications ranging from basic algebra to advanced engineering and physics problems. Mastering these techniques ensures a solid foundation for tackling more complex mathematical challenges.
Therefore, the correct answer is D. 4 and -6.
