Solve The Equation X^2 - 10x + 26 = 8 By Completing The Square And Find The Values Of A And B In The Solutions X = A - \sqrt{b} And X = A + \sqrt{b}.
In this article, we will delve into the process of solving quadratic equations by completing the square, a powerful algebraic technique. We will take a step-by-step approach to solve the given equation:
x^2 - 10x + 26 = 8
and express the solutions in the form:
x = a - \sqrt{b}
x = a + \sqrt{b}
where a and b are constants that we need to determine. Completing the square is a fundamental method in algebra, providing a way to rewrite any quadratic equation into a form that allows for easy isolation of the variable. This technique is not only crucial for solving equations but also for understanding the structure and properties of quadratic functions, such as finding the vertex of a parabola or expressing the quadratic in vertex form.
H2: Understanding the Method of Completing the Square
Before diving into the specific problem, let's first understand the core concept behind completing the square. The goal is to transform a quadratic expression of the form x² + bx + c into a perfect square trinomial, which can be factored as (x + k)² for some constant k. This transformation makes it easier to solve for x. The key idea is to manipulate the original equation by adding and subtracting a specific value to create the perfect square. This value is determined by taking half of the coefficient of the x term and squaring it. The process allows us to rewrite the quadratic equation in a form where the variable x appears only once, making it straightforward to isolate and solve for its values.
For a general quadratic expression x² + bx, we add and subtract (b/2)² to complete the square:
x^2 + bx = x^2 + bx + (b/2)^2 - (b/2)^2 = (x + b/2)^2 - (b/2)^2
This manipulation transforms the original expression into a perfect square trinomial, (x + b/2)², plus a constant term, -( b/2)². The result is a form of the equation that is easy to solve because it involves the square of a binomial. This method is particularly useful when the quadratic equation cannot be easily factored using simple factorization techniques. It provides a systematic approach to finding the roots of any quadratic equation, regardless of the complexity of its coefficients. Moreover, understanding the process of completing the square lays the foundation for more advanced algebraic concepts and techniques.
H2: Step-by-Step Solution
Let's apply the method of completing the square to the given equation:
x^2 - 10x + 26 = 8
H3: Step 1: Move the Constant Term to the Right Side
First, we need to isolate the quadratic and linear terms on the left side of the equation. To do this, subtract 26 from both sides:
x^2 - 10x = 8 - 26
x^2 - 10x = -18
This step is crucial because it sets the stage for completing the square. By moving the constant term to the right side, we create space on the left side to add the necessary term to form a perfect square trinomial. This isolates the terms containing x and prepares the equation for the next steps in the process. Ensuring that the constant term is on the opposite side of the equation is a fundamental step in solving quadratic equations by completing the square.
H3: Step 2: Complete the Square
Now, we complete the square for the left side of the equation. The coefficient of our x term is -10. We take half of this coefficient, which is -5, and square it: (-5)² = 25. We add this value to both sides of the equation:
x^2 - 10x + 25 = -18 + 25
x^2 - 10x + 25 = 7
The rationale behind this step is to transform the left side of the equation into a perfect square trinomial, which can then be factored into the square of a binomial. By adding the same value to both sides of the equation, we maintain the equality while simultaneously creating the desired perfect square on the left side. This step is the heart of the completing the square method and is essential for rewriting the equation in a solvable form.
H3: Step 3: Factor the Left Side
The left side is now a perfect square trinomial. We can factor it as:
(x - 5)^2 = 7
This step simplifies the equation significantly. The perfect square trinomial x² - 10x + 25 is the result of squaring the binomial (x - 5). By factoring the left side, we consolidate the x terms into a single squared expression, making it easier to isolate x. This transformation is crucial because it allows us to take the square root of both sides of the equation in the next step, which will lead us closer to the solutions for x. Factoring the perfect square trinomial is a key step in the completing the square process.
H3: Step 4: Take the Square Root of Both Sides
Take the square root of both sides of the equation:
\sqrt{(x - 5)^2} = \pm\sqrt{7}
x - 5 = \pm\sqrt{7}
Remember to include both the positive and negative square roots. Taking the square root of both sides is a fundamental operation in solving equations involving squares. It allows us to undo the squaring operation and move closer to isolating the variable x. The inclusion of both the positive and negative square roots is essential because both values, when squared, will yield the original value inside the square root. This step acknowledges the two possible solutions for x and ensures that we capture both roots of the quadratic equation.
H3: Step 5: Isolate x
Add 5 to both sides to solve for x:
x = 5 \pm \sqrt{7}
This step isolates x and gives us the solutions to the quadratic equation. By adding 5 to both sides, we undo the subtraction that was previously applied to x, leaving x alone on one side of the equation. The result is two distinct solutions, one with the addition of the square root and one with the subtraction of the square root. These solutions represent the roots of the quadratic equation and the points where the parabola intersects the x-axis. Isolating x is the final algebraic manipulation needed to obtain the explicit solutions to the equation.
H2: Final Answer
The solutions are in the form:
x = a - \sqrt{b}
x = a + \sqrt{b}
Comparing this with our solution:
x = 5 - \sqrt{7}
x = 5 + \sqrt{7}
We can see that a = 5 and b = 7.
Therefore, the values are:
a = 5
b = 7
By comparing the solutions obtained through the completing the square method with the desired form x = a ± √b, we can directly identify the values of a and b. In this case, a corresponds to the constant term that was added to both sides of the equation when isolating x, and b corresponds to the constant term under the square root. This final step provides a clear and concise answer to the problem, expressing the solutions in the specified format.
H2: Conclusion
In this article, we have successfully solved the quadratic equation x² - 10x + 26 = 8 by completing the square. This method allowed us to rewrite the equation in a form that is easily solvable, leading us to the solutions x = 5 - √7 and x = 5 + √7. We found that a = 5 and b = 7, completing the solution as requested. Mastering completing the square is an invaluable skill in algebra, providing a reliable method for solving quadratic equations and laying the groundwork for more advanced mathematical concepts.
The process of completing the square not only allows us to solve quadratic equations but also provides deeper insights into the structure and properties of quadratic functions. It is a fundamental technique that is used extensively in various areas of mathematics and its applications. By understanding and practicing this method, students can develop a strong foundation in algebra and enhance their problem-solving skills.