Solving Quadratic Equations: Find Solutions For Z² + 10z - 24 = 0

Introduction to Quadratic Equations

Let's dive into the fascinating world of quadratic equations! In this article, we'll break down how to solve the equation z² + 10z - 24 = 0. Quadratic equations are polynomial equations of the second degree, meaning they have a term with the variable raised to the power of 2. They often appear in various fields of mathematics and science, making it essential to understand how to solve them. We will explore different methods to find the solutions, also known as roots or zeros, of this specific equation. By the end of this guide, you’ll have a clear understanding of how to tackle similar problems and confidently find the values of z that satisfy the equation.

Understanding the Quadratic Equation

Before we jump into solving, it’s important to understand the standard form of a quadratic equation, which is ax² + bx + c = 0. In our case, the equation is z² + 10z - 24 = 0. Here, a is the coefficient of the z² term, b is the coefficient of the z term, and c is the constant term. For our equation, a = 1, b = 10, and c = -24. Recognizing these coefficients is the first step in choosing the appropriate method to solve the equation. There are several ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages, and the best approach often depends on the specific equation. For instance, factoring is efficient when the equation can be easily factored, while the quadratic formula always works, regardless of the complexity of the equation. In the following sections, we’ll explore factoring as a primary method for solving z² + 10z - 24 = 0 because it provides a straightforward and intuitive way to find the solutions.

Method 1: Factoring the Quadratic Equation

One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. To factor the equation z² + 10z - 24 = 0, we need to find two numbers that multiply to -24 (the constant term) and add up to 10 (the coefficient of the z term). Let's think about the factors of -24. Possible pairs are (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), and (-4, 6). Among these pairs, -2 and 12 satisfy both conditions: -2 * 12 = -24 and -2 + 12 = 10. Therefore, we can rewrite the equation as (z - 2)(z + 12) = 0. Now, to find the solutions for z, we set each factor equal to zero. This is based on the principle that if the product of two factors is zero, then at least one of the factors must be zero. So, we have two equations: z - 2 = 0 and z + 12 = 0. Solving these linear equations is straightforward. For z - 2 = 0, we add 2 to both sides, giving us z = 2. For z + 12 = 0, we subtract 12 from both sides, resulting in z = -12. Thus, the solutions to the quadratic equation z² + 10z - 24 = 0 are z = 2 and z = -12. Factoring is a powerful technique because it breaks down a complex equation into simpler parts, making it easier to find the solutions. It's also a great way to deepen your understanding of algebraic manipulations and equation-solving strategies.

Step-by-Step Factoring Process

Let's break down the factoring process step-by-step to ensure you grasp each part. First, we identify the coefficients in our equation z² + 10z - 24 = 0. As mentioned earlier, a = 1, b = 10, and c = -24. The core of factoring involves finding two numbers that meet specific criteria related to these coefficients. The first criterion is that the two numbers must multiply to the constant term, c, which is -24 in our case. The second criterion is that these two numbers must add up to the coefficient of the z term, b, which is 10. Systematically listing the factor pairs of -24 helps in this step. We consider both positive and negative pairs to ensure we cover all possibilities. After listing the pairs, such as (1, -24), (-1, 24), (2, -12), (-2, 12), and so on, we check each pair to see if they sum up to 10. As we found earlier, the pair (-2, 12) fits the criteria because -2 * 12 = -24 and -2 + 12 = 10. Once we have these numbers, we can rewrite the quadratic equation in factored form. The general form for factoring a quadratic equation z² + bz + c = 0 when a = 1 is (z + number 1)(z + number 2) = 0. Applying this to our equation, we get (z - 2)(z + 12) = 0. The final step is setting each factor equal to zero and solving for z. This gives us the solutions z = 2 and z = -12. By following these steps methodically, you can confidently factor a wide range of quadratic equations and find their solutions.

Method 2: Using the Quadratic Formula

Another powerful method for solving quadratic equations is using the quadratic formula. This method is particularly useful when the equation is difficult or impossible to factor. The quadratic formula is given by: z = [-b ± √(b² - 4ac)] / (2a). Here, a, b, and c are the coefficients from the quadratic equation in the standard form ax² + bx + c = 0. In our equation, z² + 10z - 24 = 0, we have a = 1, b = 10, and c = -24. To apply the formula, we substitute these values into the equation. First, let's calculate the discriminant, which is the part under the square root: b² - 4ac. In our case, this is 10² - 4 * 1 * (-24) = 100 + 96 = 196. Now, we can plug this into the quadratic formula: z = [-10 ± √196] / (2 * 1). The square root of 196 is 14, so we have z = [-10 ± 14] / 2. This gives us two possible solutions. The first solution is z = (-10 + 14) / 2 = 4 / 2 = 2. The second solution is z = (-10 - 14) / 2 = -24 / 2 = -12. Therefore, the solutions to the equation z² + 10z - 24 = 0, using the quadratic formula, are z = 2 and z = -12. The quadratic formula is a versatile tool that works for any quadratic equation, making it an essential technique in algebra. It ensures you can find solutions even when factoring is not straightforward.

Applying the Quadratic Formula Step-by-Step

To ensure you're comfortable using the quadratic formula, let's go through the process step by step with our equation, z² + 10z - 24 = 0. The first step is to correctly identify the coefficients a, b, and c. In this case, a = 1, b = 10, and c = -24. Next, we write down the quadratic formula: z = [-b ± √(b² - 4ac)] / (2a). It’s helpful to have this formula memorized, but if not, make sure you have it handy. Now, we substitute the values of a, b, and c into the formula. This gives us z = [-10 ± √(10² - 4 * 1 * (-24))] / (2 * 1). The next step is to simplify the expression inside the square root, which is the discriminant. We calculate 10² - 4 * 1 * (-24) = 100 + 96 = 196. So, our equation becomes z = [-10 ± √196] / 2. We then find the square root of 196, which is 14. Our equation now simplifies to z = [-10 ± 14] / 2. This leads us to two possible solutions because of the ± sign. We calculate these separately. For the first solution, we use the + sign: z = (-10 + 14) / 2 = 4 / 2 = 2. For the second solution, we use the - sign: z = (-10 - 14) / 2 = -24 / 2 = -12. Thus, the solutions are z = 2 and z = -12. By breaking the process into these steps, you can apply the quadratic formula accurately and confidently, even for more complex quadratic equations. This method provides a reliable way to find solutions, regardless of the equation's factorability.

Verifying the Solutions

After finding the solutions to the quadratic equation, it’s crucial to verify them. Verifying ensures that the values we’ve calculated are indeed correct and satisfy the original equation. This step is a great way to catch any potential errors made during the solving process. To verify the solutions z = 2 and z = -12 for the equation z² + 10z - 24 = 0, we substitute each value back into the equation and check if the equation holds true. Let's start with z = 2. Substituting into the equation, we get (2)² + 10(2) - 24 = 4 + 20 - 24 = 24 - 24 = 0. Since the result is 0, z = 2 is indeed a solution. Now, let's verify z = -12. Substituting into the equation, we get (-12)² + 10(-12) - 24 = 144 - 120 - 24 = 144 - 144 = 0. Again, the result is 0, confirming that z = -12 is also a solution. By substituting the solutions back into the original equation and confirming that they make the equation true, we increase our confidence in the accuracy of our results. This verification step is a fundamental practice in mathematics and helps build a strong understanding of equation-solving techniques.

Importance of Checking Your Answers

Checking your answers, as we've just demonstrated, is a vital part of the problem-solving process in mathematics. This step not only confirms the correctness of your solutions but also enhances your understanding of the problem and the methods used to solve it. In the context of quadratic equations, verifying your solutions ensures that the values you've found truly satisfy the equation. This is particularly important because mistakes can occur during factoring, applying the quadratic formula, or completing the square. By substituting the solutions back into the original equation, you can catch arithmetic errors or incorrect applications of formulas. Moreover, checking your answers reinforces the concept of what a solution actually means. A solution to a quadratic equation is a value that, when substituted for the variable, makes the equation true. Verifying helps solidify this understanding. Additionally, the habit of checking your work is a valuable skill that extends beyond mathematics. In any field that involves problem-solving, verifying results helps ensure accuracy and reliability. Whether you're solving a simple algebraic equation or a complex engineering problem, taking the time to check your work is an investment in the correctness of your results.

Conclusion

In this article, we explored methods for solving the quadratic equation z² + 10z - 24 = 0. We looked at factoring and using the quadratic formula, both of which led us to the solutions z = 2 and z = -12. We also emphasized the importance of verifying these solutions to ensure their accuracy. Mastering quadratic equations is a fundamental skill in algebra and provides a solid foundation for more advanced mathematical concepts. Whether you prefer factoring for its simplicity or the quadratic formula for its versatility, understanding these methods will empower you to tackle a wide range of problems. Remember, practice is key to proficiency, so try solving various quadratic equations to build your skills and confidence. Happy solving!

For further learning on quadratic equations and other mathematical topics, you may find helpful resources at Khan Academy.