Solving Quadratic Equations By Factoring Z^2 - Z - 6 = 0

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Factoring quadratic equations is a fundamental skill in algebra, and it's crucial for solving a wide range of mathematical problems. In this article, we will delve into the process of solving the quadratic equation z2−z−6=0z^2 - z - 6 = 0 by factoring. This comprehensive guide will not only provide a step-by-step solution but also offer valuable insights into the underlying concepts and techniques involved in factoring quadratic equations. By understanding these principles, you will be well-equipped to tackle similar problems with confidence and precision. Whether you're a student looking to improve your algebra skills or simply someone interested in mathematics, this article will serve as a valuable resource. We'll break down each step, explain the reasoning behind it, and provide clear examples to ensure you grasp the concepts fully. So, let's embark on this mathematical journey and master the art of solving quadratic equations by factoring!

Understanding Quadratic Equations

Before we dive into the specifics of solving z2−z−6=0z^2 - z - 6 = 0, let's take a moment to understand what quadratic equations are and why they are so important in mathematics. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (in this case, 'z') is 2. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The 'x' represents the variable we are trying to solve for.

Why are quadratic equations important? They arise in numerous real-world applications, from physics and engineering to economics and finance. For example, quadratic equations can be used to model the trajectory of a projectile, the shape of a satellite dish, or the growth of a population. Their ubiquity in various fields makes it essential to understand how to solve them efficiently.

There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. Factoring, the method we will focus on in this article, is often the most straightforward approach when the equation can be factored easily. It involves expressing the quadratic expression as a product of two linear expressions. By setting each of these linear expressions equal to zero, we can find the solutions (or roots) of the quadratic equation. Understanding the underlying principles of factoring is not only crucial for solving quadratic equations but also for simplifying algebraic expressions and solving other types of equations. So, let's proceed with the specific steps involved in factoring z2−z−6=0z^2 - z - 6 = 0.

Step-by-Step Solution: Factoring z2−z−6=0z^2 - z - 6 = 0

Now, let's get to the heart of the matter and solve the equation z2−z−6=0z^2 - z - 6 = 0 by factoring. Factoring involves expressing the quadratic expression as a product of two binomials. Here's a detailed breakdown of the steps involved:

1. Identify the Coefficients

First, we need to identify the coefficients 'a', 'b', and 'c' in our equation. In the equation z2−z−6=0z^2 - z - 6 = 0, we have:

  • a = 1 (the coefficient of z2z^2)
  • b = -1 (the coefficient of z)
  • c = -6 (the constant term)

These coefficients will guide us in the factoring process. Understanding the values of 'a', 'b', and 'c' is crucial because they dictate the structure of the binomials we are trying to find. The value of 'a' tells us the leading coefficient of the quadratic, 'b' is the coefficient of the linear term, and 'c' is the constant term. Each of these values plays a specific role in determining how the quadratic factors. For instance, 'c' determines the product of the constant terms in the binomials, while 'b' influences the sum of those terms.

2. Find Two Numbers That Multiply to 'c' and Add Up to 'b'

This is the core of the factoring process. We need to find two numbers that satisfy two conditions:

  • Their product is equal to 'c' (-6 in this case).
  • Their sum is equal to 'b' (-1 in this case).

Let's think about the factors of -6. We have the following pairs:

  • 1 and -6
  • -1 and 6
  • 2 and -3
  • -2 and 3

Now, we need to check which of these pairs adds up to -1. The pair 2 and -3 satisfies this condition because 2 + (-3) = -1. This step is often the most challenging part of factoring for beginners. It requires a bit of trial and error, but with practice, you'll develop a knack for identifying the correct pair of numbers quickly. A systematic approach, such as listing out the factors of 'c' and then checking their sums, can be very helpful. Remember, the signs of the numbers are crucial, so pay close attention to whether the product and sum need to be positive or negative.

3. Rewrite the Middle Term

Now that we have found the two numbers (2 and -3), we rewrite the middle term (-z) using these numbers. So, we rewrite the equation as:

z2+2z−3z−6=0z^2 + 2z - 3z - 6 = 0

By rewriting the middle term, we've essentially split the linear term into two parts, each involving one of the numbers we found in the previous step. This manipulation is crucial because it sets us up for the next step: factoring by grouping. Rewriting the middle term doesn't change the value of the expression; it merely changes its appearance. This is a common technique in algebra, and it's used in various contexts, not just factoring quadratics. The key is to choose the right numbers to split the middle term so that the subsequent factoring by grouping is possible.

4. Factor by Grouping

Next, we factor by grouping. We group the first two terms and the last two terms:

(z2+2z)+(−3z−6)=0(z^2 + 2z) + (-3z - 6) = 0

Now, we factor out the greatest common factor (GCF) from each group:

z(z + 2) - 3(z + 2) = 0

Notice that we now have a common factor of (z + 2) in both terms. Factoring by grouping is a powerful technique that allows us to simplify expressions with four terms into a product of two binomials. The key is to choose the grouping carefully so that a common factor emerges. In this case, grouping the first two terms and the last two terms allowed us to factor out 'z' from the first group and '-3' from the second group, resulting in the common factor of (z + 2). If we hadn't chosen the correct pair of numbers in step 2, this step might not have been possible.

5. Factor Out the Common Binomial

We factor out the common binomial (z + 2):

(z + 2)(z - 3) = 0

We have successfully factored the quadratic expression into two binomials. This step is the culmination of the factoring process. We've transformed the original quadratic expression into a product of two linear expressions. This form is incredibly useful because it allows us to find the solutions of the equation easily. The factored form reveals the roots of the equation, which are the values of 'z' that make the equation true. Factoring out the common binomial is a crucial step because it simplifies the equation into a form where we can apply the zero-product property, which is the key to finding the solutions.

6. Apply the Zero-Product Property

The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have:

(z + 2)(z - 3) = 0

So, either (z + 2) = 0 or (z - 3) = 0.

The zero-product property is a fundamental principle in algebra, and it's the reason why factoring is such a powerful method for solving equations. It allows us to break down a complex equation into simpler equations that we can solve individually. The property is based on the fact that zero multiplied by any number is zero. Therefore, if a product of factors equals zero, at least one of the factors must be zero. This property is not only applicable to quadratic equations but also to other types of equations involving products of expressions.

7. Solve for z

Now, we solve each equation separately:

  • z + 2 = 0 => z = -2
  • z - 3 = 0 => z = 3

Therefore, the solutions to the equation z2−z−6=0z^2 - z - 6 = 0 are z = -2 and z = 3. These values are the roots of the quadratic equation, and they represent the points where the parabola defined by the equation intersects the x-axis. Solving for 'z' in this step involves isolating the variable in each linear equation. This is a straightforward process that typically involves adding or subtracting a constant from both sides of the equation. The solutions we obtain are the values of 'z' that satisfy the original quadratic equation, meaning that if we substitute these values back into the equation, it will hold true. This is a good way to check our work and ensure that we have factored the equation correctly.

Final Answer

The solutions to the equation z2−z−6=0z^2 - z - 6 = 0 are:

D. z = -2 or z = 3

These solutions represent the values of 'z' that make the equation true. We have successfully solved the equation by factoring, demonstrating a powerful algebraic technique. Understanding the steps involved in factoring quadratic equations is crucial for solving a wide range of mathematical problems. From identifying the coefficients to applying the zero-product property, each step plays a vital role in finding the solutions. By mastering this technique, you'll be well-equipped to tackle similar problems with confidence and accuracy. Remember, practice is key to improving your factoring skills, so try solving various quadratic equations using this method. The more you practice, the more comfortable and proficient you'll become.

Practice Problems

To solidify your understanding of factoring quadratic equations, try solving the following practice problems:

  1. x2+5x+6=0x^2 + 5x + 6 = 0
  2. y2−4y+3=0y^2 - 4y + 3 = 0
  3. 2a2+7a+3=02a^2 + 7a + 3 = 0

These problems will give you an opportunity to apply the steps we've discussed in this article and reinforce your factoring skills. Working through these exercises will help you internalize the process and identify any areas where you may need further practice. Remember to follow the steps carefully, starting with identifying the coefficients, finding the appropriate numbers, rewriting the middle term, factoring by grouping, and applying the zero-product property. The solutions to these problems can be found online or in most algebra textbooks. Don't hesitate to seek help from a teacher or tutor if you encounter any difficulties. Practice is essential for mastering any mathematical skill, and factoring quadratic equations is no exception.

Conclusion

In this article, we've explored the process of solving the equation z2−z−6=0z^2 - z - 6 = 0 by factoring. We've broken down each step in detail, from identifying the coefficients to applying the zero-product property. We've also discussed the importance of quadratic equations and the various methods for solving them. Factoring is a powerful technique that allows us to express a quadratic expression as a product of two binomials, making it easier to find the solutions of the equation. By understanding the underlying principles of factoring and practicing regularly, you can develop the skills necessary to solve a wide range of quadratic equations. Remember, mathematics is a skill that improves with practice, so don't be discouraged if you encounter challenges along the way. Keep practicing, keep learning, and you'll be well on your way to mastering algebra. Factoring quadratic equations is a fundamental skill that will serve you well in your mathematical journey, so make sure to invest the time and effort to develop a strong understanding of this technique.