Solving X In The Quadratic Equation X^2 + 10x + 12 = 36 A Step By Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of x in the quadratic equation x² + 10x + 12 = 36. Quadratic equations, characterized by the presence of a squared term (x²), often present a unique challenge compared to linear equations. However, with a systematic approach, we can unravel the mysteries of these equations and arrive at the solutions.
Understanding Quadratic Equations
Before we dive into the solution, let's take a moment to understand what quadratic equations are and why they hold such significance in mathematics. 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 general form of a quadratic equation is expressed as:
ax² + bx + c = 0
where a, b, and c are constants, and a ≠0. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions represent the points where the parabola, the graphical representation of the quadratic equation, intersects the x-axis. Quadratic equations have wide-ranging applications in various fields, including physics, engineering, economics, and computer science. They are used to model projectile motion, optimize designs, and analyze financial trends, highlighting their importance in both theoretical and practical contexts. Understanding how to solve quadratic equations is therefore crucial for anyone pursuing studies or careers in these areas. The ability to manipulate and solve these equations allows for a deeper understanding of the relationships between variables and the systems they represent. Mastery of these skills not only enhances problem-solving abilities but also provides a foundation for more advanced mathematical concepts. In this article, we will explore a step-by-step approach to solving the given quadratic equation, ensuring a clear understanding of each step involved. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar quadratic equations effectively.
Step 1: Rearranging the Equation
The first step in solving the equation x² + 10x + 12 = 36 is to rearrange it into the standard quadratic form, which is ax² + bx + c = 0. This form is essential because it allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. To achieve this standard form, we need to move all the terms to one side of the equation, leaving zero on the other side. In our case, we have the equation x² + 10x + 12 = 36. To get zero on the right side, we subtract 36 from both sides of the equation. This ensures that we maintain the equality of the equation, as whatever operation we perform on one side, we must also perform on the other. Subtracting 36 from both sides, we get:
x² + 10x + 12 - 36 = 36 - 36
Simplifying the equation, we combine the constant terms 12 and -36, which gives us -24. So, the equation becomes:
x² + 10x - 24 = 0
Now, the equation is in the standard quadratic form, where a = 1, b = 10, and c = -24. This form makes it easier to identify the coefficients, which are crucial for applying different solution methods. The process of rearranging the equation into standard form is a fundamental step in solving quadratic equations. It sets the stage for the subsequent steps and allows us to systematically apply the appropriate techniques. Without this step, it would be significantly more challenging to solve the equation. By ensuring that the equation is in standard form, we can accurately identify the coefficients and proceed with the chosen solution method, whether it's factoring, completing the square, or using the quadratic formula. This methodical approach is key to successfully solving quadratic equations.
Step 2: Factoring the Quadratic Equation
Factoring is a method used to solve quadratic equations by expressing the quadratic expression as a product of two linear factors. This method is particularly effective when the quadratic equation can be easily factored, which means finding two binomials that, when multiplied together, give the original quadratic expression. In our case, we have the quadratic equation x² + 10x - 24 = 0. To factor this equation, we need to find two numbers that multiply to give the constant term (-24) and add up to the coefficient of the x term (10). Let's consider the factors of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), and (-4, 6). Among these pairs, the pair (-2, 12) satisfies our conditions because -2 multiplied by 12 equals -24, and -2 plus 12 equals 10. Therefore, we can rewrite the quadratic equation as:
(x - 2)(x + 12) = 0
This factorization means that the product of the two binomials (x - 2) and (x + 12) is equal to zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for x. So, we have two equations:
x - 2 = 0
x + 12 = 0
Solving the first equation, x - 2 = 0, we add 2 to both sides, which gives us x = 2. Solving the second equation, x + 12 = 0, we subtract 12 from both sides, which gives us x = -12. Thus, the solutions to the quadratic equation x² + 10x - 24 = 0 are x = 2 and x = -12. Factoring is a powerful technique for solving quadratic equations, especially when the factors are integers. It simplifies the process of finding the roots by breaking down the quadratic expression into manageable linear factors. This method not only provides the solutions but also offers insights into the structure of the quadratic equation. By understanding the factors, we can better visualize the behavior of the quadratic function and its relationship to the x-axis.
Step 3: Solving for x
Now that we have factored the quadratic equation into (x - 2)(x + 12) = 0, we can proceed to find the values of x that satisfy the equation. As we discussed in the previous step, the zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is the key to solving for x in our factored equation. We have two factors: (x - 2) and (x + 12). Setting each factor equal to zero, we create two separate linear equations:
- x - 2 = 0
- x + 12 = 0
Let's solve each equation individually. For the first equation, x - 2 = 0, we want to isolate x on one side of the equation. To do this, we add 2 to both sides of the equation:
x - 2 + 2 = 0 + 2
This simplifies to:
x = 2
So, one solution to the quadratic equation is x = 2. Now, let's solve the second equation, x + 12 = 0. Similarly, we want to isolate x. To do this, we subtract 12 from both sides of the equation:
x + 12 - 12 = 0 - 12
This simplifies to:
x = -12
So, the second solution to the quadratic equation is x = -12. Therefore, the solutions to the quadratic equation x² + 10x - 24 = 0 are x = 2 and x = -12. These values of x are the roots or zeros of the quadratic equation. They represent the points where the parabola corresponding to the equation intersects the x-axis. Solving for x using the zero-product property is a straightforward and efficient method when the quadratic equation can be easily factored. It allows us to break down the problem into smaller, more manageable linear equations, making the solution process clear and concise. By understanding this method, you can confidently solve a wide range of quadratic equations.
Alternative Methods for Solving Quadratic Equations
While factoring is a useful method for solving quadratic equations, it is not always the most efficient or even possible method for every equation. There are other techniques available that can be used to find the solutions, also known as roots or zeros, of quadratic equations. Two commonly used alternative methods are completing the square and the quadratic formula. Let's explore each of these methods in detail.
Completing the Square
Completing the square is a method that involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly helpful when the quadratic equation cannot be easily factored. The general idea behind completing the square is to manipulate the equation so that one side is a perfect square trinomial, which is a trinomial that can be expressed as the square of a binomial. To complete the square, we start with the standard form of the quadratic equation, ax² + bx + c = 0. If a is not equal to 1, we divide the entire equation by a to make the coefficient of x² equal to 1. Then, we move the constant term (c) to the right side of the equation. Next, we take half of the coefficient of the x term (b), square it, and add it to both sides of the equation. This step is crucial because it creates the perfect square trinomial on the left side. The left side can then be factored as a binomial squared, and we can solve for x by taking the square root of both sides. For example, let's consider the equation x² + 6x + 5 = 0. To complete the square, we first move the constant term to the right side: x² + 6x = -5. Then, we take half of the coefficient of x (which is 6), square it (3² = 9), and add it to both sides: x² + 6x + 9 = -5 + 9. This simplifies to (x + 3)² = 4. Taking the square root of both sides gives us x + 3 = ±2. Solving for x, we get x = -1 and x = -5. Completing the square is a powerful method that can be used to solve any quadratic equation, regardless of whether it can be factored easily. It provides a systematic approach to finding the solutions and is particularly useful in situations where factoring is not straightforward.
The Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations. It provides a direct solution for x in terms of the coefficients a, b, and c of the quadratic equation ax² + bx + c = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
The formula may look intimidating at first, but it is a straightforward application of the coefficients of the quadratic equation. To use the quadratic formula, we simply identify the values of a, b, and c from the equation and substitute them into the formula. The ± sign indicates that there are two possible solutions for x, one with the plus sign and one with the minus sign. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the solutions. If the discriminant is positive, there are two distinct real solutions. If the discriminant is zero, there is exactly one real solution (a repeated root). If the discriminant is negative, there are two complex solutions. For example, let's solve the equation 2x² - 5x + 3 = 0 using the quadratic formula. Here, a = 2, b = -5, and c = 3. Substituting these values into the formula, we get:
x = (5 ± √((-5)² - 4 * 2 * 3)) / (2 * 2)
Simplifying, we get:
x = (5 ± √(25 - 24)) / 4
x = (5 ± √1) / 4
x = (5 ± 1) / 4
So, the two solutions are x = (5 + 1) / 4 = 1.5 and x = (5 - 1) / 4 = 1. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity. It is particularly useful when the equation cannot be easily factored or when the solutions are not rational numbers. By understanding and applying the quadratic formula, you can confidently solve a wide range of quadratic equations and gain a deeper understanding of their properties.
Conclusion
In this article, we have explored a step-by-step approach to solving the quadratic equation x² + 10x + 12 = 36. We began by rearranging the equation into the standard quadratic form, ax² + bx + c = 0, which is a crucial step for applying various solution methods. We then factored the quadratic equation, which allowed us to use the zero-product property to find the solutions. The solutions we found were x = 2 and x = -12. Additionally, we discussed alternative methods for solving quadratic equations, including completing the square and using the quadratic formula. These methods provide versatile tools for solving quadratic equations, especially when factoring is not straightforward. Completing the square involves transforming the equation into a perfect square trinomial, while the quadratic formula provides a direct solution in terms of the coefficients of the equation. By understanding and mastering these methods, you can confidently tackle a wide range of quadratic equations and gain a deeper appreciation for their mathematical properties. Quadratic equations are fundamental in various fields, including mathematics, physics, engineering, and economics. Their solutions represent critical points in modeling real-world phenomena, such as projectile motion, optimization problems, and financial analysis. The ability to solve quadratic equations efficiently and accurately is therefore an essential skill for anyone pursuing studies or careers in these areas. Moreover, the process of solving quadratic equations enhances problem-solving abilities and cultivates a deeper understanding of algebraic concepts. Whether you choose to factor, complete the square, or use the quadratic formula, the key is to approach the problem systematically and apply the appropriate techniques. With practice and perseverance, you can become proficient in solving quadratic equations and unlock their potential in various applications.
By mastering these techniques, you will be well-equipped to solve a wide range of quadratic equations and apply them to real-world problems.
