What Is The Solution Set Of The Equation 6x² - 18x - 18 = 6?
Navigating the realm of quadratic equations often involves finding the solution set, which represents the values of the variable that satisfy the equation. In this comprehensive exploration, we will delve into the process of determining the solution set for the equation 6x² - 18x - 18 = 6. By employing algebraic techniques and a step-by-step approach, we will unveil the roots of this equation and gain a deeper understanding of quadratic equations.
Transforming the Equation to Standard Form
The first crucial step in solving a quadratic equation is to transform it into the standard form, which is expressed as ax² + bx + c = 0, where a, b, and c are constants. This form allows us to readily apply various methods for finding the solutions. To achieve this, we begin by subtracting 6 from both sides of the given equation:
6x² - 18x - 18 - 6 = 6 - 6
This simplifies to:
6x² - 18x - 24 = 0
Now, we observe that all the coefficients in the equation are divisible by 6. Dividing both sides of the equation by 6, we obtain a simplified form:
x² - 3x - 4 = 0
This simplified equation is now in the standard form, making it easier to solve.
Factoring the Quadratic Expression
One of the most common methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear factors. To factor the expression x² - 3x - 4, we seek two numbers that multiply to -4 and add up to -3. These numbers are -4 and 1.
Therefore, we can rewrite the quadratic expression as:
(x - 4)(x + 1) = 0
This factorization represents the original quadratic expression in an equivalent form.
Applying 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. Applying this property to our factored equation, we have:
x - 4 = 0 or x + 1 = 0
Solving each of these linear equations for x, we get:
x = 4 or x = -1
These values of x are the solutions to the quadratic equation.
Identifying the Solution Set
The solution set of an equation is the set of all values that satisfy the equation. In this case, the solution set for the equation 6x² - 18x - 18 = 6 is {4, -1}.
Therefore, the correct answer is not provided in the options. The correct solution set is {4, -1}.
Alternative Methods for Solving Quadratic Equations
While factoring is a powerful method, it may not always be straightforward or applicable. In such cases, alternative methods can be employed to solve quadratic equations. Let's explore two such methods:
The Quadratic Formula
The quadratic formula is a universal tool for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Applying this formula to our equation x² - 3x - 4 = 0, where a = 1, b = -3, and c = -4, we get:
x = (3 ± √((-3)² - 4 * 1 * -4)) / 2 * 1
Simplifying this expression, we obtain:
x = (3 ± √(9 + 16)) / 2
x = (3 ± √25) / 2
x = (3 ± 5) / 2
This yields two solutions:
x = (3 + 5) / 2 = 4
x = (3 - 5) / 2 = -1
These solutions match the ones we obtained through factoring.
Completing the Square
Completing the square is another method that can be used to solve quadratic equations. This method involves manipulating the equation to create a perfect square trinomial on one side. To apply this method to our equation x² - 3x - 4 = 0, we first move the constant term to the right side:
x² - 3x = 4
Next, we take half of the coefficient of the x term (-3), square it ((-3/2)² = 9/4), and add it to both sides of the equation:
x² - 3x + 9/4 = 4 + 9/4
This transforms the left side into a perfect square trinomial:
(x - 3/2)² = 25/4
Taking the square root of both sides, we get:
x - 3/2 = ±√(25/4)
x - 3/2 = ±5/2
Solving for x, we obtain:
x = 3/2 ± 5/2
This gives us the two solutions:
x = (3/2 + 5/2) = 4
x = (3/2 - 5/2) = -1
Again, these solutions coincide with those obtained through factoring and the quadratic formula.
Conclusion
In this exploration, we have successfully determined the solution set of the quadratic equation 6x² - 18x - 18 = 6. By transforming the equation to standard form, factoring, and applying the zero-product property, we found the solutions to be x = 4 and x = -1. We also explored alternative methods for solving quadratic equations, including the quadratic formula and completing the square, which yielded the same solutions. Understanding these methods empowers us to tackle a wide range of quadratic equations and solve them effectively. Remember, the solution set represents the values that satisfy the equation, providing valuable insights into the behavior of quadratic functions. Therefore, the solution set for the equation 6x² - 18x - 18 = 6 is {4, -1}, which was not provided in the original options.
In this article, we'll break down the process of finding the solution set for the quadratic equation 6x² - 18x - 18 = 6. We'll walk through each step clearly, making it easy to understand how to arrive at the correct answer. Quadratic equations are a fundamental concept in algebra, and mastering them is essential for further mathematical studies. This article aims to provide a thorough explanation and ensure you grasp the concepts involved.
Step 1: Transforming the Equation into Standard Form
The first and most crucial step is to rewrite the equation in its standard quadratic form. The standard form is represented as ax² + bx + c = 0, where a, b, and c are constants. This form is necessary because it allows us to apply various methods such as factoring, using the quadratic formula, or completing the square.
To get our equation into standard form, we need to move all terms to one side, leaving zero on the other side. We start with the given equation:
6x² - 18x - 18 = 6
Subtract 6 from both sides to set the equation equal to zero:
6x² - 18x - 18 - 6 = 6 - 6
This simplifies to:
6x² - 18x - 24 = 0
Now, observe that all coefficients (6, -18, and -24) are divisible by 6. Dividing the entire equation by 6 simplifies it further, making it easier to work with:
(6x² - 18x - 24) / 6 = 0 / 6
Which gives us:
x² - 3x - 4 = 0
This simplified equation is now in standard form, which makes it ready for the next steps.
Why is Standard Form Important?
Putting a quadratic equation in standard form is important because it allows us to clearly identify the coefficients a, b, and c. These coefficients are crucial when using methods like the quadratic formula or completing the square. Furthermore, factoring often becomes easier when the equation is in standard form, as it helps in identifying the appropriate factors.
Step 2: Solving by Factoring
Factoring is a method of breaking down a quadratic expression into two binomial expressions that, when multiplied together, give the original quadratic expression. For our simplified equation, x² - 3x - 4 = 0, we need to find two numbers that multiply to -4 (the constant term) and add up to -3 (the coefficient of the x term).
Let's think of pairs of factors of -4:
- -4 and 1
- -1 and 4
- -2 and 2
Among these pairs, -4 and 1 add up to -3, which is what we need. So, we can rewrite the quadratic expression using these numbers:
(x - 4)(x + 1) = 0
When these two binomials are multiplied, they give us x² - 3x - 4. Factoring the quadratic expression allows us to use the zero-product property, which is our next step.
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 other words, if AB = 0, then either A = 0 or B = 0 (or both). This property is the key to finding the solutions once we have factored the quadratic equation.
Step 3: Applying the Zero-Product Property
Now that we have factored the equation as (x - 4)(x + 1) = 0, we can apply the zero-product property. This means we set each factor equal to zero and solve for x:
- x - 4 = 0
- x + 1 = 0
Solving the first equation:
x - 4 = 0
Add 4 to both sides:
x = 4
Solving the second equation:
x + 1 = 0
Subtract 1 from both sides:
x = -1
So, we have found two solutions for x: 4 and -1. These are the roots of the quadratic equation.
Step 4: Verifying the Solutions
To ensure that our solutions are correct, we can substitute each value of x back into the original equation and check if it holds true. Let's verify x = 4:
Original equation: 6x² - 18x - 18 = 6
Substitute x = 4:
6(4)² - 18(4) - 18 = 6
6(16) - 72 - 18 = 6
96 - 72 - 18 = 6
6 = 6
The equation holds true for x = 4.
Now, let's verify x = -1:
Original equation: 6x² - 18x - 18 = 6
Substitute x = -1:
6(-1)² - 18(-1) - 18 = 6
6(1) + 18 - 18 = 6
6 = 6
The equation also holds true for x = -1.
Since both values satisfy the original equation, our solutions are correct.
Step 5: Stating the Solution Set
The solution set is the set of all values of x that satisfy the equation. In this case, we have found two solutions: x = 4 and x = -1. Therefore, the solution set is {4, -1}.
Conclusion
In summary, to find the solution set of the quadratic equation 6x² - 18x - 18 = 6, we followed these steps:
- Transformed the equation into standard form: x² - 3x - 4 = 0
- Factored the quadratic expression: (x - 4)(x + 1) = 0
- Applied the zero-product property: x - 4 = 0 or x + 1 = 0
- Solved for x: x = 4 and x = -1
- Verified the solutions by substituting them back into the original equation.
- Stated the solution set: {4, -1}
This detailed walkthrough should provide a clear understanding of how to solve quadratic equations by factoring. Mastering these steps will help you confidently solve similar problems in the future. Remember, practice is key, so try solving more quadratic equations to reinforce your understanding.
This in-depth article will meticulously explore the process of determining the solution set, or roots, for the quadratic equation 6x² - 18x - 18 = 6. We will dissect the equation, applying algebraic techniques to unravel its solutions. Understanding how to solve quadratic equations is fundamental in mathematics, and this guide aims to provide a clear, step-by-step approach. Whether you are a student learning algebra or someone looking to refresh your skills, this comprehensive exploration will walk you through each necessary step.
Initial Equation and the Importance of Simplification
Our journey begins with the given quadratic equation:
6x² - 18x - 18 = 6
The initial step in tackling any quadratic equation is to simplify it and bring it to a more manageable form. Simplification not only makes the equation easier to work with but also reduces the likelihood of errors in subsequent steps. In this case, we will start by moving all terms to one side of the equation to set it equal to zero. This is a critical step as it transforms the equation into the standard form necessary for many solution methods.
Step 1: Setting the Equation to Zero
To set the equation to zero, we need to subtract 6 from both sides of the equation:
6x² - 18x - 18 - 6 = 6 - 6
This operation results in:
6x² - 18x - 24 = 0
Now, the equation is in a form where all terms are on one side, and the other side is zero. This is a significant milestone, as it sets the stage for further simplification and the application of various solution techniques. The next step involves identifying common factors and further simplifying the equation.
Step 2: Identifying and Dividing by Common Factors
Observing the coefficients of the quadratic equation 6x² - 18x - 24 = 0, we notice that each coefficient (6, -18, and -24) is divisible by 6. Dividing the entire equation by a common factor simplifies the equation without changing its solutions. This is an important step, especially when dealing with large coefficients, as it reduces the complexity of the numbers involved and makes the equation easier to handle.
Dividing both sides of the equation by 6, we get:
(6x² - 18x - 24) / 6 = 0 / 6
This simplifies to:
x² - 3x - 4 = 0
Now, the equation is in its simplest standard form, ax² + bx + c = 0, where a = 1, b = -3, and c = -4. This form is ideal for applying various solution methods, such as factoring, using the quadratic formula, or completing the square. In the subsequent sections, we will explore one of the most common methods for solving quadratic equations: factoring.
Solving the Simplified Quadratic Equation by Factoring
Factoring is a powerful method for solving quadratic equations, particularly when the equation can be expressed as a product of two binomials. The goal of factoring is to rewrite the quadratic expression as a product of two linear factors. In the case of our simplified equation, x² - 3x - 4 = 0, we seek two numbers that multiply to give the constant term (-4) and add up to give the coefficient of the linear term (-3).
Step 3: Factoring the Quadratic Expression
To factor the expression x² - 3x - 4, we need to identify two numbers that meet the criteria mentioned above. Let's consider pairs of factors of -4:
- -4 and 1
- -1 and 4
- -2 and 2
Among these pairs, -4 and 1 satisfy our conditions because (-4) * (1) = -4 and (-4) + (1) = -3. Therefore, we can rewrite the quadratic expression as:
(x - 4)(x + 1)
Setting this factored expression equal to zero, we have:
(x - 4)(x + 1) = 0
This factored form of the equation allows us to use the zero-product property, which is a cornerstone in solving factored equations. The next step involves applying this property to find the values of x that satisfy the equation.
Step 4: Applying 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 two factors, (x - 4) and (x + 1), and their product is equal to zero. Applying the zero-product property, we set each factor equal to zero and solve for x:
- x - 4 = 0
- x + 1 = 0
Solving the first equation, x - 4 = 0, we add 4 to both sides:
x = 4
Solving the second equation, x + 1 = 0, we subtract 1 from both sides:
x = -1
Thus, we have found two potential solutions for x: 4 and -1. These values are the roots of the quadratic equation. To ensure that these solutions are correct, it is essential to verify them by substituting them back into the original equation.
Verification and the Final Solution Set
Verifying the solutions is a crucial step in solving any equation, as it confirms that the values we have found indeed satisfy the original equation. In the context of quadratic equations, verification helps to avoid errors that may arise during the simplification or factoring process. We will now verify our solutions, x = 4 and x = -1, by substituting them into the original equation, 6x² - 18x - 18 = 6.
Step 5: Verifying the Solutions
Let's start by verifying x = 4:
6(4)² - 18(4) - 18 = 6
6(16) - 72 - 18 = 6
96 - 72 - 18 = 6
6 = 6
The equation holds true for x = 4. Now, let's verify x = -1:
6(-1)² - 18(-1) - 18 = 6
6(1) + 18 - 18 = 6
6 = 6
The equation also holds true for x = -1. Since both values satisfy the original equation, our solutions are correct.
Step 6: Stating the Solution Set
The solution set of a quadratic equation is the set of all values of x that satisfy the equation. In our case, we have found two solutions: x = 4 and x = -1. Therefore, the solution set is {4, -1}.
Alternative Methods and Conclusion
While factoring is an effective method for solving quadratic equations, it is not always straightforward or applicable. In cases where factoring is difficult or impossible, alternative methods such as the quadratic formula or completing the square can be used. These methods are particularly useful for equations with irrational or complex roots. However, for our equation, factoring provided a direct and efficient way to find the solutions.
Conclusion
In conclusion, we have successfully determined the solution set for the quadratic equation 6x² - 18x - 18 = 6. By simplifying the equation, factoring, applying the zero-product property, and verifying our solutions, we found the roots to be x = 4 and x = -1. This comprehensive exploration has demonstrated the step-by-step process involved in solving quadratic equations by factoring, providing a clear understanding of the techniques and principles involved. The solution set for the equation is {4, -1}, highlighting the importance of systematic problem-solving in mathematics.