Solutions For X² + 6x - 6 = 10 A Comprehensive Guide
Understanding how to solve quadratic equations is a fundamental skill in algebra. In this article, we will delve into the process of solving the quadratic equation x² + 6x - 6 = 10. We'll explore different methods and techniques to find the solutions, also known as the roots, of this equation. This exploration will not only help you solve this specific problem but also equip you with the knowledge to tackle other quadratic equations you might encounter.
Transforming the Equation into Standard Form
Before we can apply any solution methods, the first crucial step is to transform the given equation into the standard quadratic form, which is ax² + bx + c = 0. This form allows us to easily identify the coefficients a, b, and c, which are essential for various solution techniques. In our case, the equation is x² + 6x - 6 = 10. To bring it to the standard form, we need to subtract 10 from both sides of the equation. This gives us:
x² + 6x - 6 - 10 = 10 - 10
Simplifying this, we get:
x² + 6x - 16 = 0
Now, our equation is in the standard quadratic form, where a = 1, b = 6, and c = -16. With the equation in this form, we can proceed to explore different methods for finding its solutions. The standard form not only provides a clear structure but also makes the equation amenable to techniques like factoring, completing the square, and the quadratic formula. Understanding this transformation is the foundation for solving any quadratic equation effectively.
Method 1: Factoring the Quadratic Equation
Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a product of two binomials. The factoring method relies on the principle that if the product of two factors is zero, then at least one of the factors must be zero. In our equation, x² + 6x - 16 = 0, we need to find two numbers that multiply to -16 (the constant term) and add up to 6 (the coefficient of the x term). These two numbers are 8 and -2, since 8 * -2 = -16 and 8 + (-2) = 6.
Therefore, we can rewrite the quadratic equation as:
(x + 8)(x - 2) = 0
Now, applying the principle mentioned earlier, we set each factor equal to zero and solve for x:
x + 8 = 0 or x - 2 = 0
Solving these linear equations, we get:
x = -8 or x = 2
Thus, the solutions to the quadratic equation x² + 6x - 16 = 0 are x = -8 and x = 2. Factoring is often the quickest method when it is applicable, as it avoids the more complex calculations involved in other methods like the quadratic formula or completing the square. However, it's important to note that not all quadratic equations can be easily factored, so it's essential to be familiar with other solution techniques as well.
Method 2: Applying the Quadratic Formula
When factoring isn't straightforward or possible, the quadratic formula provides a reliable method for solving any quadratic equation in the standard form ax² + bx + c = 0. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, x² + 6x - 16 = 0, we have a = 1, b = 6, and c = -16. Substituting these values into the quadratic formula, we get:
x = (-6 ± √(6² - 4 * 1 * -16)) / (2 * 1)
Simplifying the expression under the square root:
x = (-6 ± √(36 + 64)) / 2
x = (-6 ± √100) / 2
x = (-6 ± 10) / 2
Now, we have two possible solutions:
x₁ = (-6 + 10) / 2 = 4 / 2 = 2
x₂ = (-6 - 10) / 2 = -16 / 2 = -8
Therefore, the solutions to the quadratic equation, as found using the quadratic formula, are x = 2 and x = -8. The quadratic formula is a versatile tool that guarantees a solution for any quadratic equation, making it an indispensable technique in algebra. It's particularly useful when the roots are not easily discernible through factoring or when dealing with equations that have irrational or complex roots.
Method 3: Completing the Square
Completing the square is another powerful method for solving quadratic equations. This technique involves manipulating the equation to create a perfect square trinomial on one side. Starting with our equation, x² + 6x - 16 = 0, we first move the constant term to the right side:
x² + 6x = 16
Next, we need to add a value to both sides of the equation to complete the square. This value is determined by taking half of the coefficient of the x term (which is 6), squaring it, and adding the result to both sides. Half of 6 is 3, and 3 squared is 9. So, we add 9 to both sides:
x² + 6x + 9 = 16 + 9
Now, the left side is a perfect square trinomial, which can be factored as:
(x + 3)² = 25
To solve for x, we take the square root of both sides:
√(x + 3)² = ±√25
x + 3 = ±5
Now, we have two equations to solve:
x + 3 = 5 or x + 3 = -5
Solving for x in each case, we get:
x = 5 - 3 = 2
x = -5 - 3 = -8
Thus, the solutions to the quadratic equation x² + 6x - 16 = 0, as found by completing the square, are x = 2 and x = -8. Completing the square is a valuable method not only for solving quadratic equations but also for understanding the structure of quadratic expressions and their graphs. It's particularly useful in situations where we need to rewrite the quadratic expression in vertex form, which is essential for identifying the vertex of the parabola.
Verifying the Solutions
After finding the solutions to a quadratic equation, it's always a good practice to verify them by substituting them back into the original equation. This step helps ensure that the solutions are correct and that no algebraic errors were made during the solving process. Our original equation, after transforming it to the standard form, is x² + 6x - 16 = 0. We found the solutions to be x = 2 and x = -8.
Let's first verify x = 2:
(2)² + 6(2) - 16 = 0
4 + 12 - 16 = 0
16 - 16 = 0
0 = 0
This confirms that x = 2 is indeed a solution.
Now, let's verify x = -8:
(-8)² + 6(-8) - 16 = 0
64 - 48 - 16 = 0
16 - 16 = 0
0 = 0
This confirms that x = -8 is also a solution. Since both solutions satisfy the original equation, we can be confident in our answers. Verification is a crucial step in problem-solving, especially in mathematics, as it helps catch mistakes and build confidence in the results.
Conclusion
In conclusion, we have explored various methods for solving the quadratic equation x² + 6x - 6 = 10, including factoring, using the quadratic formula, and completing the square. All three methods led us to the same solutions: x = 2 and x = -8. We also emphasized the importance of transforming the equation into the standard form ax² + bx + c = 0 before applying any solution method and verifying the solutions to ensure their correctness. Mastering these techniques equips you with the skills to solve a wide range of quadratic equations and lays a strong foundation for more advanced mathematical concepts. Understanding the different approaches to solving quadratic equations allows for flexibility and adaptability in problem-solving, as some methods may be more efficient than others depending on the specific equation. The ability to confidently solve quadratic equations is a valuable asset in mathematics and its applications.
The correct answer is D. x = -8 or x = 2.
