Solving $x^2 + 10x + 12 = 36$ A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving a quadratic equation, specifically the equation $x^2 + 10x + 12 = 36$. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and economics. They often arise when modeling real-world phenomena such as projectile motion, optimization problems, and financial analysis. Mastering the techniques for solving quadratic equations is therefore essential for anyone pursuing studies or careers in these areas. This article will provide a comprehensive, step-by-step guide to solving the given equation, ensuring clarity and understanding for readers of all backgrounds.

Our main goal here is to find the values of $x$ that satisfy the given equation. These values are known as the roots or solutions of the equation. To accomplish this, we will employ a combination of algebraic manipulations and the application of the quadratic formula. We will begin by transforming the equation into its standard form, which is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. This form allows us to readily identify the coefficients necessary for applying the quadratic formula. Following the transformation, we will carefully calculate the discriminant, a key component of the quadratic formula that provides insights into the nature of the roots. The discriminant will tell us whether the equation has two distinct real roots, one repeated real root, or two complex roots. Once we have computed the discriminant, we will proceed to apply the quadratic formula, substituting the appropriate values to obtain the roots of the equation. Each step of the process will be explained in detail, with clear justifications and examples, to ensure that you not only understand the method but also its underlying principles. By the end of this article, you will have a solid grasp of how to solve quadratic equations and will be equipped to tackle similar problems with confidence.

Step 1: Transforming the Equation to Standard Form

The first crucial step in solving the quadratic equation $x^2 + 10x + 12 = 36$ is to transform it into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form is essential because it allows us to easily identify the coefficients $a$, $b$, and $c$, which are necessary for applying the quadratic formula and other solution methods. To achieve this standard form, we need to move all terms to one side of the equation, leaving zero on the other side. In our case, this involves subtracting 36 from both sides of the equation.

Starting with the given equation, $x^2 + 10x + 12 = 36$, we subtract 36 from both sides to eliminate the constant term on the right side. This operation maintains the equality of the equation, as we are performing the same operation on both sides. The subtraction yields: $x^2 + 10x + 12 - 36 = 36 - 36$. Simplifying this, we combine the constant terms on the left side: $x^2 + 10x - 24 = 0$. Now, the equation is in the standard quadratic form, where $a = 1$, $b = 10$, and $c = -24$. Identifying these coefficients is a critical step, as they will be directly used in the subsequent application of the quadratic formula. This transformation not only simplifies the equation but also sets the stage for a systematic solution process. By ensuring that the equation is in standard form, we can apply various methods, including factoring, completing the square, or the quadratic formula, with greater ease and accuracy. The careful execution of this step is fundamental to the successful resolution of the quadratic equation.

Step 2: Applying the Quadratic Formula

Now that we have transformed the equation into its standard form, $x^2 + 10x - 24 = 0$, we can proceed with applying the quadratic formula. The quadratic formula is a powerful tool for finding the roots of any quadratic equation in the form $ax^2 + bx + c = 0$. It states that the solutions for $x$ are given by: x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}. This formula provides a direct method for calculating the roots, regardless of whether the equation can be easily factored. To use the formula, we first need to identify the coefficients $a$, $b$, and $c$ from our standard form equation. As we determined earlier, in our case, $a = 1$, $b = 10$, and $c = -24$. These values will be substituted into the quadratic formula to compute the solutions for $x$.

Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get: x = rac{-10 ext{±} ext{√}(10^2 - 4(1)(-24))}{2(1)}. Now, we need to simplify this expression step by step. First, let's simplify the term inside the square root: $10^2 = 100$ and $4(1)(-24) = -96$. So, the expression becomes: x = rac{-10 ext{±} ext{√}(100 - (-96))}{2}. Next, we simplify the expression inside the square root further: $100 - (-96) = 100 + 96 = 196$. Thus, we have: x = rac{-10 ext{±} ext{√}196}{2}. The square root of 196 is 14, so the equation simplifies to: x = rac{-10 ext{±} 14}{2}. Now, we have two separate solutions to calculate, one with the plus sign and one with the minus sign. These calculations will give us the two roots of the quadratic equation. The careful and methodical application of the quadratic formula is essential to ensure the accuracy of the solutions. By substituting the coefficients and simplifying the expression step by step, we can systematically find the roots of the equation.

Step 3: Calculating the Two Possible Solutions

Having simplified the quadratic formula to x = rac{-10 ext{±} 14}{2}, we now need to calculate the two possible solutions for $x$. This involves considering both the addition and subtraction cases of the $ ext{±}$ symbol. The quadratic formula inherently provides two solutions because quadratic equations, being second-degree polynomials, can have up to two distinct roots. These roots represent the values of $x$ that satisfy the equation, and they are the points where the parabola defined by the quadratic equation intersects the x-axis. To find these roots, we will first compute the solution using the plus sign and then the solution using the minus sign.

First, let's calculate the solution using the plus sign: x = rac{-10 + 14}{2}. Simplifying the numerator, we have $-10 + 14 = 4$. So, the equation becomes x = rac{4}{2}, which simplifies to $x = 2$. This is one of the roots of the quadratic equation. Next, we calculate the solution using the minus sign: x = rac{-10 - 14}{2}. Simplifying the numerator, we have $-10 - 14 = -24$. So, the equation becomes x = rac{-24}{2}, which simplifies to $x = -12$. This is the second root of the quadratic equation. Therefore, the two solutions for $x$ are $x = 2$ and $x = -12$. These values are the roots of the equation $x^2 + 10x - 24 = 0$, and they represent the points where the parabola intersects the x-axis. By systematically considering both the addition and subtraction cases in the quadratic formula, we have successfully found both solutions to the equation. These solutions are critical for understanding the behavior of the quadratic function and for solving related problems in various fields.

Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 + 10x + 12 = 36$ by following a systematic, step-by-step approach. Our journey began with transforming the equation into its standard form, $x^2 + 10x - 24 = 0$, which is a crucial preparatory step for applying the quadratic formula. By subtracting 36 from both sides of the original equation, we were able to rearrange it into the standard form, where the coefficients $a$, $b$, and $c$ could be easily identified. These coefficients are the building blocks for the quadratic formula, which is a powerful tool for finding the roots of any quadratic equation.

Next, we applied the quadratic formula, x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}, substituting the values $a = 1$, $b = 10$, and $c = -24$. This step involved careful arithmetic and algebraic manipulation to ensure accuracy. We simplified the expression step by step, first by calculating the discriminant, which is the term inside the square root, and then by taking the square root. This process led us to the expression x = rac{-10 ext{±} 14}{2}, which represents the two possible solutions for $x$. Finally, we calculated the two solutions by considering both the addition and subtraction cases of the $ ext{±}$ symbol. This gave us the roots $x = 2$ and $x = -12$. These roots are the values of $x$ that satisfy the original equation, and they represent the points where the parabola defined by the quadratic equation intersects the x-axis.

The ability to solve quadratic equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering the techniques outlined in this article, you are well-equipped to tackle similar problems and to apply these skills in more advanced contexts. The step-by-step approach, from transforming the equation to applying the quadratic formula and calculating the roots, provides a clear and structured method for solving quadratic equations. Understanding this process not only enhances your problem-solving abilities but also deepens your understanding of mathematical principles. Therefore, the solutions to the equation $x^2 + 10x + 12 = 36$ are $x = 2$ and $x = -12$, and the correct answer is C. $x = -12$ or $x = 2$.