Solving $x^2 + X - 12 = 0$ Find The Value Of $x + 4$ Given $x > 0$

In this article, we will delve into the solution of the quadratic equation $x^2 + x - 12 = 0$. We will explore the methods to find the roots of this equation and, given the condition that $x > 0$, determine the value of $x + 4$. This is a fundamental problem in algebra, often encountered in high school mathematics and various competitive exams. Understanding how to solve quadratic equations is crucial for a strong foundation in mathematics. We will break down the process step-by-step, ensuring clarity and a comprehensive understanding of the concepts involved.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a ≠ 0$. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of $x$ that satisfy the equation. Solving quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and economics. Mastering the techniques to solve these equations is essential for anyone pursuing advanced studies in these areas.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its advantages and applications. The most common methods include factoring, completing the square, and using the quadratic formula. Each method provides a unique approach to finding the roots of the equation. Understanding these different methods allows you to choose the most efficient approach based on the specific equation you are dealing with. Let's explore these methods in more detail:

1. Factoring

Factoring is a method used when the quadratic expression can be factored into two binomials. This method involves finding two numbers that multiply to the constant term ($c$) and add up to the coefficient of the linear term ($b$). Once the quadratic expression is factored, we can set each factor equal to zero and solve for $x$. Factoring is often the quickest method when the quadratic equation has integer roots. For instance, in our equation $x^2 + x - 12 = 0$, we can look for two numbers that multiply to -12 and add up to 1. This method leverages the reverse process of expanding two binomials, making it an efficient technique when applicable.

2. Completing the Square

Completing the square is a method that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. This method involves adding and subtracting a constant term to both sides of the equation to create a perfect square. Completing the square is particularly useful when the quadratic equation cannot be easily factored. This method provides a systematic way to solve any quadratic equation, even those with complex roots. The process involves manipulating the equation to isolate the variable term and then adding a specific constant to both sides to complete the square, making it a versatile technique.

3. Quadratic Formula

The quadratic formula is a general formula that provides the solutions to any quadratic equation. The formula is given by:

$$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it can be factored or not. It is especially useful when the roots are not rational numbers. This formula is derived from the method of completing the square and provides a direct way to calculate the roots, making it an indispensable tool in algebra.

Solving $x^2 + x - 12 = 0$ Using Factoring

Let's solve the quadratic equation $x^2 + x - 12 = 0$ using the factoring method. As mentioned earlier, we need to find two numbers that multiply to -12 and add up to 1. These numbers are 4 and -3. Therefore, we can rewrite the equation as:

$$x^2 + 4x - 3x - 12 = 0$$

Now, we can factor by grouping:

$$x(x + 4) - 3(x + 4) = 0$$

$$(x - 3)(x + 4) = 0$$

Setting each factor equal to zero gives us the solutions:

$$x - 3 = 0 \implies x = 3$$

$$x + 4 = 0 \implies x = -4$$

So, the solutions to the equation are $x = 3$ and $x = -4$. Factoring is an efficient method for solving quadratic equations when the coefficients allow for easy identification of the factors. This method relies on the distributive property of multiplication over addition and provides a straightforward way to find the roots of the equation.

Identifying the Positive Solution

The problem states that $x > 0$, so we choose the positive solution, which is $x = 3$. The condition $x > 0$ narrows down our solutions to only the positive root. This is a common practice in mathematical problems where additional constraints are given to ensure a unique solution or to reflect real-world scenarios where negative values may not be applicable. Selecting the appropriate solution based on the given conditions is a critical step in problem-solving.

Finding the Value of $x + 4$

Now that we have found the value of $x$, we can calculate $x + 4$:

$$x + 4 = 3 + 4 = 7$$

Thus, the value of $x + 4$ is 7. This final step involves a simple arithmetic operation, but it is an essential part of the problem. It demonstrates the importance of carefully reading the question and providing the exact answer requested. The ability to perform basic arithmetic operations accurately is crucial for success in mathematics.

Conclusion

In conclusion, by solving the quadratic equation $x^2 + x - 12 = 0$ using factoring and considering the condition $x > 0$, we found that $x = 3$. Therefore, the value of $x + 4$ is 7. This problem illustrates the importance of understanding quadratic equations and the various methods to solve them. Mastering these skills is essential for success in algebra and beyond. The ability to solve quadratic equations is a cornerstone of mathematical education and has wide-ranging applications in various fields.

This article has provided a step-by-step guide to solving the given quadratic equation, emphasizing the importance of each step and the underlying mathematical principles. By understanding these concepts, students can confidently tackle similar problems and build a strong foundation in algebra. The methods discussed here are not only applicable to academic settings but also to real-world problem-solving scenarios, highlighting the practical relevance of mathematics.