Solving The Quadratic Equation W^2 + 2w - 24 = 0

Introduction to Quadratic Equations

In the realm of mathematics, quadratic equations hold a prominent position due to their frequent appearance in various applications and theoretical contexts. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, represent the values of the variable (x in the general form) that satisfy the equation. Solving quadratic equations is a fundamental skill in algebra, and several methods exist to find these solutions. Among the common methods are factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations.

Our focus in this article is on solving the specific quadratic equation w² + 2w - 24 = 0. This equation is a classic example of a quadratic equation that can be solved using multiple approaches, including factoring. We will primarily explore the method of factoring to find the solutions, but it is worth noting that the quadratic formula could also be applied to arrive at the same result. Understanding how to solve quadratic equations is crucial for various fields, including physics, engineering, economics, and computer science. The ability to find the roots of a quadratic equation allows us to model and solve problems related to projectile motion, optimization, and curve fitting, among other applications.

Before diving into the solution, it is essential to recognize the structure of the given equation. The equation w² + 2w - 24 = 0 is in the standard quadratic form, where a = 1, b = 2, and c = -24. Identifying these coefficients is a key step in choosing the appropriate method for solving the equation. In this case, the equation is factorable, which makes factoring an efficient and straightforward method for finding the solutions. As we proceed, we will demonstrate the steps involved in factoring the equation and determining the values of w that make the equation true. This process not only provides the solutions but also enhances our understanding of the relationship between the coefficients and the roots of a quadratic equation.

Factoring the Quadratic Equation

The equation we aim to solve is w² + 2w - 24 = 0. To solve this equation by factoring, our initial goal is to rewrite the quadratic expression as a product of two binomials. This involves finding two numbers that satisfy two specific conditions: their product must equal the constant term (-24 in this case), and their sum must equal the coefficient of the linear term (2 in this case). By identifying these two numbers, we can decompose the quadratic expression into a form that allows us to easily find the roots.

Let's systematically search for these two numbers. We need two integers that multiply to -24 and add up to 2. We can start by listing the factor pairs of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), and (-4, 6). Among these pairs, the pair (-4, 6) meets our criteria. When multiplied, -4 and 6 give -24, and when added, they give 2. This is exactly what we need to factor the quadratic expression successfully.

Now that we have identified the two numbers, we can rewrite the middle term of the quadratic equation using these numbers. Specifically, we replace 2w with -4w + 6w. This gives us the expression w² - 4w + 6w - 24 = 0. The next step involves grouping the terms in pairs and factoring out the greatest common factor (GCF) from each pair. From the first two terms, w² - 4w, we can factor out w, which yields w(w - 4). From the last two terms, 6w - 24, we can factor out 6, resulting in 6(w - 4). Now, we have w(w - 4) + 6(w - 4) = 0.

Notice that both terms now have a common factor of (w - 4). We can factor out this common binomial factor, which gives us (w - 4)(w + 6) = 0. This is the factored form of the original quadratic equation. The equation is now expressed as a product of two factors 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 w. This critical step transforms the problem of solving a quadratic equation into solving two simpler linear equations.

Solving for the Roots

After successfully factoring the quadratic equation w² + 2w - 24 = 0 into the form (w - 4)(w + 6) = 0, the next crucial step is to apply the zero-product property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In our case, the two factors are (w - 4) and (w + 6). Therefore, to find the values of w that satisfy the equation, we set each factor equal to zero and solve for w.

First, let's consider the factor (w - 4). Setting it equal to zero gives us the equation w - 4 = 0. To solve for w, we add 4 to both sides of the equation. This isolates w on the left side and yields the solution w = 4. This is one of the roots of the quadratic equation. It means that when w is equal to 4, the original equation w² + 2w - 24 = 0 is satisfied. We can verify this by substituting w = 4 back into the original equation: (4)² + 2(4) - 24 = 16 + 8 - 24 = 0, which confirms that w = 4 is indeed a solution.

Next, we turn our attention to the factor (w + 6). Setting this factor equal to zero gives us the equation w + 6 = 0. To solve for w, we subtract 6 from both sides of the equation. This isolates w and gives us the solution w = -6. This is the second root of the quadratic equation. Similar to the previous solution, we can verify this by substituting w = -6 back into the original equation: (-6)² + 2(-6) - 24 = 36 - 12 - 24 = 0, which confirms that w = -6 is also a solution.

Thus, by applying the zero-product property to the factored form of the quadratic equation, we have found two distinct solutions for w: w = 4 and w = -6. These values represent the points where the quadratic function w² + 2w - 24 intersects the w-axis. Understanding how to find these roots is essential for analyzing the behavior of quadratic functions and solving related problems in various fields. The two solutions, w = 4 and w = -6, correspond to option C in the provided choices, which is w = -6, w = 4.

Conclusion: Solutions and Verification

In summary, we have successfully solved the quadratic equation w² + 2w - 24 = 0 using the factoring method. This involved identifying two numbers that multiply to -24 and add up to 2, which led us to the factored form (w - 4)(w + 6) = 0. Applying the zero-product property, we set each factor equal to zero and solved for w, obtaining the solutions w = 4 and w = -6. These are the two roots of the given quadratic equation.

To ensure the accuracy of our solutions, we verified each root by substituting it back into the original equation. For w = 4, we found that (4)² + 2(4) - 24 = 16 + 8 - 24 = 0, confirming that 4 is indeed a solution. Similarly, for w = -6, we found that (-6)² + 2(-6) - 24 = 36 - 12 - 24 = 0, confirming that -6 is also a solution. The process of verification is a crucial step in problem-solving, as it helps to catch any potential errors and ensures that the obtained solutions are correct.

Therefore, the solutions to the quadratic equation w² + 2w - 24 = 0 are w = -6 and w = 4. This corresponds to option C in the provided choices. The ability to solve quadratic equations is a fundamental skill in algebra and is widely applicable in various mathematical and real-world contexts. Factoring, completing the square, and the quadratic formula are all valuable tools for finding the roots of quadratic equations. Understanding these methods and their applications is essential for students and professionals in various fields. The systematic approach we have used, from factoring to applying the zero-product property and verifying the solutions, provides a clear and reliable method for solving quadratic equations of this type. Mastering these techniques allows for confident and accurate problem-solving in a wide range of situations.