Solving X² - 8x - 6 = -6 By Factoring A Step-by-Step Guide
Solve x²-8x-6=-6 by factoring.
In the realm of algebra, quadratic equations hold a position of prominence, serving as fundamental tools for modeling various real-world phenomena. Among the techniques employed to solve these equations, factoring stands out as an elegant and efficient method. In this comprehensive guide, we delve into the intricacies of solving quadratic equations by factoring, using the specific example of x² - 8x - 6 = -6 to illustrate the step-by-step process. Our goal is to equip you with the knowledge and skills necessary to confidently tackle similar problems and gain a deeper understanding of quadratic equations.
Understanding Quadratic Equations
Before we embark on the factoring journey, it is essential to grasp the essence of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is expressed as:
ax² + bx + c = 0
where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to solve for. These constants play crucial roles in determining the characteristics of the quadratic equation and its solutions. The coefficient 'a' dictates the parabola's direction (upward if positive, downward if negative), while 'b' influences its horizontal position, and 'c' governs its vertical shift. Understanding these parameters provides valuable insights into the behavior of quadratic equations.
Factoring, as a solution technique, hinges on the principle of transforming the quadratic equation into a product of two linear expressions. This transformation allows us to leverage the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property forms the cornerstone of our approach to solving quadratic equations by factoring.
The Zero-Product Property: A Cornerstone of Factoring
The zero-product property is a fundamental concept in algebra that underpins the factoring method for solving quadratic equations. It 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. This principle can be expressed mathematically as:
If A * B = 0, then A = 0 or B = 0 (or both)
This seemingly simple property is the key to unlocking solutions through factoring. By rewriting a quadratic equation in the form of a product of linear factors, we can set each factor equal to zero and solve for the variable. This process yields the roots or solutions of the quadratic equation, which represent the values of 'x' that satisfy the equation.
For instance, consider the factored quadratic equation:
(x - 2)(x + 3) = 0
According to the zero-product property, either (x - 2) = 0 or (x + 3) = 0. Solving these linear equations individually, we find x = 2 and x = -3. These are the solutions to the quadratic equation, the points where the parabola intersects the x-axis.
The zero-product property transforms the challenge of solving a quadratic equation into a more manageable task of solving linear equations. This is the essence of factoring, a powerful technique for finding the roots of quadratic equations.
Step-by-Step Solution for x² - 8x - 6 = -6
Let's now apply the factoring method to solve the quadratic equation x² - 8x - 6 = -6. We'll break down the solution into clear, manageable steps, ensuring you grasp each concept thoroughly.
Step 1: Rearrange the Equation to Standard Form
The first crucial step in solving any quadratic equation is to rearrange it into the standard form, ax² + bx + c = 0. This form allows us to readily identify the coefficients 'a', 'b', and 'c', which are essential for factoring. In our case, we need to eliminate the constant term on the right side of the equation.
To do this, we add 6 to both sides of the equation:
x² - 8x - 6 + 6 = -6 + 6
This simplifies to:
x² - 8x = 0
Now, the equation is in the standard form, with a = 1, b = -8, and c = 0. This sets the stage for the next step, which involves factoring the quadratic expression.
Step 2: Factor the Quadratic Expression
The next step involves factoring the quadratic expression x² - 8x. Factoring is the process of breaking down an expression into a product of simpler expressions. In this case, we look for two linear expressions that, when multiplied together, give us the original quadratic expression.
To factor x² - 8x, we identify the common factor between the two terms, which is 'x'. We can then factor out 'x' from both terms:
x(x - 8) = 0
We have successfully factored the quadratic expression into a product of two linear factors: 'x' and '(x - 8)'. This transformation is the heart of the factoring method, as it allows us to apply the zero-product property.
Step 3: Apply the Zero-Product Property
With the quadratic equation factored as x(x - 8) = 0, we can now invoke the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero.
Applying this to our factored equation, we set each factor equal to zero:
x = 0 or x - 8 = 0
This step transforms the single quadratic equation into two simpler linear equations, each of which can be solved independently.
Step 4: Solve for x
We now have two simple linear equations to solve:
- x = 0
- x - 8 = 0
The first equation is already solved, giving us one solution: x = 0.
To solve the second equation, we add 8 to both sides:
x - 8 + 8 = 0 + 8
This gives us the second solution: x = 8.
Therefore, the solutions to the quadratic equation x² - 8x - 6 = -6 are x = 0 and x = 8. These are the values of 'x' that satisfy the original equation.
Verification of Solutions
To ensure the accuracy of our solutions, it is always a good practice to verify them by substituting them back into the original equation. This step helps to catch any potential errors made during the factoring or solving process.
Verification for x = 0
Substituting x = 0 into the original equation x² - 8x - 6 = -6, we get:
(0)² - 8(0) - 6 = -6
Simplifying, we have:
0 - 0 - 6 = -6
-6 = -6
The equation holds true, confirming that x = 0 is indeed a solution.
Verification for x = 8
Substituting x = 8 into the original equation x² - 8x - 6 = -6, we get:
(8)² - 8(8) - 6 = -6
Simplifying, we have:
64 - 64 - 6 = -6
-6 = -6
The equation holds true, confirming that x = 8 is also a solution.
Since both solutions satisfy the original equation, we can confidently conclude that our solutions are correct.
Alternative Methods for Solving Quadratic Equations
While factoring is a powerful technique, it is not always the most efficient method for solving quadratic equations. Some quadratic equations are difficult or impossible to factor using simple methods. In such cases, alternative techniques like the quadratic formula and completing the square come into play. Let's briefly explore these methods.
The Quadratic Formula
The quadratic formula is a universal solution for any quadratic equation, regardless of its factorability. It provides a direct way to calculate the solutions, given the coefficients 'a', 'b', and 'c' of the quadratic equation in standard form (ax² + bx + c = 0). The quadratic formula is expressed as:
x = (-b ± √(b² - 4ac)) / 2a
This formula guarantees finding the solutions, whether they are real or complex. It is a valuable tool in any algebra student's arsenal.
Completing the Square
Completing the square is another technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side, which can then be easily factored. This method is particularly useful when the quadratic equation is not readily factorable by traditional methods. Completing the square provides a systematic approach to solving quadratic equations, ensuring that solutions can be found even in challenging cases.
Conclusion: Mastering Quadratic Equations Through Factoring
In this comprehensive guide, we have explored the method of solving quadratic equations by factoring. We dissected the process into clear, manageable steps, using the example of x² - 8x - 6 = -6 to illustrate the technique. We emphasized the importance of rearranging the equation to standard form, factoring the quadratic expression, applying the zero-product property, and solving for x. Furthermore, we underscored the significance of verifying solutions to ensure accuracy.
Factoring is a powerful tool in the realm of algebra, providing an elegant and efficient way to solve quadratic equations. However, it is essential to recognize its limitations and be familiar with alternative methods like the quadratic formula and completing the square. By mastering these techniques, you can confidently tackle a wide range of quadratic equations and deepen your understanding of algebraic principles. Remember, practice is key to proficiency. The more you solve quadratic equations, the more adept you will become at identifying the most appropriate solution method and executing it effectively.
This article has equipped you with the knowledge and skills necessary to solve quadratic equations by factoring. Embrace the challenge, practice diligently, and unlock the world of quadratic equations with confidence.