Solving The Quadratic Equation X^2 + 10x + 12 = 36

Introduction

In this article, we will explore how to solve the quadratic equation $x^2 + 10x + 12 = 36$. Quadratic equations are fundamental in algebra and appear in various mathematical and real-world applications. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to 0. Solving a quadratic equation involves finding the values of $x$ that satisfy the equation, which are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations. Understanding how to solve quadratic equations is crucial for anyone studying algebra and higher-level mathematics. This article will guide you through the step-by-step process of solving the given equation, explaining each step in detail to ensure a clear understanding. We will also discuss why this approach is effective and highlight common pitfalls to avoid. By the end of this article, you will have a solid grasp of how to solve quadratic equations and be able to apply this knowledge to other similar problems. Let's dive into the process and find the solutions for $x$ in the equation $x^2 + 10x + 12 = 36$.

Step 1: Rewrite the Equation in Standard Form

The first crucial step in solving any quadratic equation is to rewrite it in the standard form, which is $ax^2 + bx + c = 0$. This form allows us to easily identify the coefficients $a$, $b$, and $c$, which are essential for applying methods like factoring, completing the square, or using the quadratic formula. In our given equation, $x^2 + 10x + 12 = 36$, we need to move the constant term from the right side to the left side to achieve the standard form. To do this, we subtract 36 from both sides of the equation. This maintains the equality and brings all terms to one side, leaving zero on the other side. The process involves a simple subtraction operation, but it's a fundamental step that sets the stage for further solution techniques. By rearranging the equation into standard form, we make it easier to apply the subsequent steps in solving for $x$. This ensures that we have a clear and organized approach to finding the solutions. Once the equation is in standard form, we can accurately identify the coefficients $a$, $b$, and $c$, which are essential for using the quadratic formula or factoring methods. This step is not just about rearranging terms; it's about preparing the equation for a systematic solution. Let's proceed with the subtraction to get the equation into its standard quadratic form and pave the way for finding the roots.

Subtracting 36 from both sides of the equation, we get:

$x^2 + 10x + 12 - 36 = 36 - 36$

Simplifying, we have:

$x^2 + 10x - 24 = 0$

Step 2: Factor the Quadratic Equation

Now that we have the quadratic equation in the standard form $x^2 + 10x - 24 = 0$, the next step is to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. This method is efficient if the quadratic expression can be easily factored. To factor the quadratic $x^2 + 10x - 24$, we look for two numbers that multiply to -24 (the constant term) and add up to 10 (the coefficient of the $x$ term). These two numbers are 12 and -2, because $12 imes -2 = -24$ and $12 + (-2) = 10$. By identifying these numbers, we can rewrite the quadratic expression as a product of two binomials, which simplifies the process of finding the solutions for $x$. Factoring is a critical technique in solving quadratic equations, and mastering it can significantly speed up the solution process. The ability to quickly identify the factors makes solving the equation much more straightforward than using alternative methods like the quadratic formula. This step is essential for understanding the structure of the quadratic expression and how it relates to the roots of the equation. Let's proceed by rewriting the quadratic equation using these factors and see how it leads us to the solutions for $x$. Understanding the relationship between the factors and the roots is a key concept in algebra, and this step illustrates that connection.

Using these numbers, we can factor the quadratic expression as:

$(x + 12)(x - 2) = 0$

Step 3: Set Each Factor Equal to Zero

After factoring the quadratic equation into $(x + 12)(x - 2) = 0$, the next crucial step is to apply the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if $AB = 0$, then either $A = 0$ or $B = 0$ (or both). This principle is fundamental in solving factored equations, as it allows us to break down the problem into simpler equations. By setting each factor equal to zero, we create two separate linear equations that are much easier to solve. This approach simplifies the process of finding the values of $x$ that satisfy the original quadratic equation. The zero-product property is a cornerstone of algebra and is widely used in various mathematical problems. Understanding and applying this property correctly is essential for solving factored equations efficiently. This step transforms a single quadratic equation into two linear equations, making the solutions much more accessible. Let's proceed by setting each factor to zero and solving for $x$ to find the roots of the equation. This technique not only simplifies the solution process but also provides a clear and logical pathway to the final answers.

To find the values of $x$, we set each factor equal to zero:

$x + 12 = 0$ or $x - 2 = 0$

Step 4: Solve for $x$

Now that we have two simple linear equations, $x + 12 = 0$ and $x - 2 = 0$, the final step is to solve each equation for $x$. This involves isolating $x$ on one side of the equation by performing basic algebraic operations. For the first equation, $x + 12 = 0$, we subtract 12 from both sides to isolate $x$. This gives us one solution for $x$. For the second equation, $x - 2 = 0$, we add 2 to both sides to isolate $x$, which gives us the second solution. Solving these linear equations is straightforward and provides the roots of the original quadratic equation. This step demonstrates the power of breaking down a complex problem into simpler parts, which is a common strategy in mathematics. By finding these solutions, we complete the process of solving the quadratic equation. Each solution represents a value of $x$ that, when substituted back into the original equation, will make the equation true. Let's complete these simple algebraic steps to find the solutions and conclude our problem-solving journey. The ability to solve linear equations is a fundamental skill in algebra, and this step reinforces its importance in the context of solving quadratic equations.

Solving the first equation:

$x + 12 = 0$ $x = -12$

Solving the second equation:

$x - 2 = 0$ $x = 2$

Conclusion

In conclusion, the solutions for the equation $x^2 + 10x + 12 = 36$ are $x = -12$ and $x = 2$. We arrived at these solutions by first rewriting the equation in standard quadratic form, then factoring the quadratic expression, and finally using the zero-product property to solve for $x$. This process demonstrates a systematic approach to solving quadratic equations, which is a fundamental skill in algebra. Understanding how to solve quadratic equations is essential for various mathematical applications and real-world problems. Each step in the solution process is crucial, from rearranging the equation to identifying the correct factors. The ability to factor quadratic expressions efficiently can significantly simplify the solution process. Furthermore, the zero-product property is a powerful tool that allows us to break down complex equations into simpler, manageable parts. By mastering these techniques, one can confidently tackle quadratic equations and related problems. The solutions we found, $x = -12$ and $x = 2$, are the values that make the original equation true. These solutions are also known as the roots or zeros of the quadratic equation. This article provided a detailed walkthrough of the solution process, highlighting each step and explaining the underlying principles. With practice, these methods can become second nature, allowing for quick and accurate solutions to quadratic equations. Remember to always check your solutions by substituting them back into the original equation to ensure their validity. This final step ensures that your answers are correct and reinforces your understanding of the solution process.

Therefore, the correct answer is:

A. $x = -12$ or $x = 2$