Solving $x^2 - 8x + 5 = 0$ By Completing The Square Method

Introduction

In the realm of mathematics, solving quadratic equations is a fundamental skill. Among various methods available, completing the square stands out as a powerful technique, offering not only solutions but also a deeper understanding of the structure of quadratic equations. In this article, we will delve into the process of solving the quadratic equation $x^2 - 8x + 5 = 0$ using the completing the square method. This method, while sometimes perceived as intricate, provides a systematic approach to rewriting quadratic equations in a form that readily reveals their solutions. Understanding this technique is crucial for anyone studying algebra, calculus, and related fields. It’s not just about finding the roots; it’s about grasping the underlying principles that govern quadratic expressions. This article will guide you through each step with detailed explanations, ensuring that you not only understand the solution to this specific equation but also the broader application of the completing the square method.

Understanding the Completing the Square Method

The completing the square method is a technique used to rewrite a quadratic equation in the form $(x - h)^2 + k = 0$, where $h$ and $k$ are constants. This form is particularly useful because it allows us to easily identify the vertex of the parabola represented by the quadratic equation, and more importantly, it simplifies the process of finding the roots. The core idea behind this method is to manipulate the quadratic expression by adding and subtracting a specific value to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as $(x + a)^2$ or $(x - a)^2$. This manipulation makes the equation easier to solve because taking the square root of both sides allows us to isolate $x$. The method is especially valuable when the quadratic equation cannot be easily factored using traditional methods. Furthermore, completing the square is the foundation for deriving the quadratic formula, making it a cornerstone technique in algebra. By mastering this method, you gain a deeper insight into the nature of quadratic equations and their graphical representations.

The General Process of Completing the Square

The general process involves several key steps. First, ensure that the coefficient of the $x^2$ term is 1. If it is not, divide the entire equation by this coefficient. Next, move the constant term to the right side of the equation. This sets the stage for creating the perfect square trinomial on the left side. The critical step is to take half of the coefficient of the $x$ term, square it, and add it to both sides of the equation. This ensures that the left side becomes a perfect square without changing the equation's balance. Once the perfect square trinomial is formed, it can be factored into the square of a binomial. The equation is now in the form $(x - h)^2 = -k$, where $h$ and $k$ are constants. Taking the square root of both sides isolates $(x - h)$, leading to two possible solutions for $x$. Remember to consider both the positive and negative square roots. Finally, solve for $x$ by adding $h$ to both sides. This systematic approach ensures accurate and efficient solutions, highlighting the power and versatility of the completing the square method. By understanding each step, you can confidently apply this technique to a wide range of quadratic equations.

Step-by-Step Solution for $x^2 - 8x + 5 = 0$

Let's apply the completing the square method to the given equation $x^2 - 8x + 5 = 0$. We will proceed step-by-step to ensure clarity and understanding.

Step 1: Ensure the Coefficient of $x^2$ is 1

In our equation, $x^2 - 8x + 5 = 0$, the coefficient of the $x^2$ term is already 1. This simplifies our initial step, as no division is required. If the coefficient were not 1, we would divide the entire equation by that coefficient to proceed. This preliminary check is crucial because completing the square works most directly when the leading coefficient is 1. Starting with this condition allows us to focus on the core steps of the method without the added complexity of dealing with a non-unity leading coefficient. This sets a solid foundation for the subsequent steps, ensuring a smoother and more accurate solution process.

Step 2: Move the Constant Term to the Right Side

Next, we move the constant term, which is 5, to the right side of the equation. To do this, we subtract 5 from both sides of the equation:

$x^2 - 8x + 5 - 5 = 0 - 5$

This simplifies to:

$x^2 - 8x = -5$

Moving the constant term is a critical step because it isolates the terms containing $x$ on one side, preparing the equation for the creation of a perfect square trinomial. This separation allows us to focus solely on the $x^2$ and $x$ terms when completing the square, making the process more manageable. By isolating these terms, we set the stage for the next crucial step: adding a specific value to both sides of the equation to complete the square.

Step 3: Complete the Square

Now comes the core of the completing the square method. We need to add a value to both sides of the equation to make the left side a perfect square trinomial. To find this value, we take half of the coefficient of the $x$ term, which is -8, and square it.

Half of -8 is -4, and $(-4)^2 = 16$. So, we add 16 to both sides of the equation:

$x^2 - 8x + 16 = -5 + 16$

This simplifies to:

$x^2 - 8x + 16 = 11$

Adding 16 completes the square on the left side, transforming the quadratic expression into a perfect square trinomial. This is the pivotal step in the method, as it sets up the equation for easy factoring and subsequent solution. The careful calculation of the value to add ensures that the left side can be expressed as the square of a binomial, which significantly simplifies the process of finding the roots.

Step 4: Factor the Left Side as a Perfect Square

The left side of the equation, $x^2 - 8x + 16$, is now a perfect square trinomial. It can be factored as:

$(x - 4)^2 = 11$

Factoring the perfect square trinomial is a crucial step in simplifying the equation. By recognizing that $x^2 - 8x + 16$ is equivalent to $(x - 4)^2$, we transform the equation into a more manageable form. This step allows us to express the quadratic in a way that directly relates to its roots. The binomial square form is essential for the next step, which involves taking the square root of both sides to isolate $x$. This factorization highlights the elegance of the completing the square method, demonstrating how a seemingly complex quadratic expression can be simplified to reveal its underlying structure.

Step 5: Take the Square Root of Both Sides

To solve for $x$, we take the square root of both sides of the equation:

$\sqrt{(x - 4)^2} = \pm\sqrt{11}$

This gives us:

$x - 4 = \pm\sqrt{11}$

Taking the square root of both sides is a fundamental step in isolating $x$. It’s important to remember to include both the positive and negative square roots, as both values satisfy the original equation. This consideration of both roots is crucial for finding all possible solutions to the quadratic equation. The $\pm$ symbol indicates that we have two separate equations to solve, leading to two distinct values for $x$. This step directly follows from the factored form and brings us closer to the final solutions by undoing the squaring operation.

Step 6: Solve for $x$

Finally, we solve for $x$ by adding 4 to both sides of the equation:

$x = 4 \pm \sqrt{11}$

This gives us two solutions:

$x = 4 + \sqrt{11}$ and $x = 4 - \sqrt{11}$

Solving for $x$ by adding 4 to both sides isolates the variable and provides the two roots of the quadratic equation. These solutions represent the points where the parabola intersects the x-axis. The two distinct values, $4 + \sqrt{11}$ and $4 - \sqrt{11}$, demonstrate the nature of quadratic equations, which typically have two solutions. This final step completes the process of solving the equation using the completing the square method, showcasing the power and precision of this technique in finding the roots of quadratic expressions.

Conclusion

In this comprehensive guide, we have successfully solved the quadratic equation $x^2 - 8x + 5 = 0$ using the completing the square method. By following each step meticulously, we transformed the equation into a form that readily revealed its solutions. The steps included ensuring the coefficient of $x^2$ was 1, moving the constant term, completing the square by adding the appropriate value to both sides, factoring the perfect square trinomial, taking the square root of both sides, and finally, solving for $x$. The two solutions we found are $x = 4 + \sqrt{11}$ and $x = 4 - \sqrt{11}$. This method not only provides the solutions but also enhances our understanding of quadratic equations and their properties. Completing the square is a valuable tool in algebra, offering a systematic approach to solving quadratics and forming the basis for other important concepts, such as the quadratic formula. Mastering this technique will undoubtedly strengthen your mathematical skills and problem-solving abilities.

Practice Problems

To solidify your understanding of the completing the square method, try solving the following quadratic equations:

1. $x^2 + 6x - 7 = 0$
2. $x^2 - 4x + 1 = 0$
3. $2x^2 + 8x - 10 = 0$ (Hint: Divide by 2 first)

Working through these practice problems will reinforce the steps involved in completing the square and help you become more confident in applying this method to various quadratic equations. Remember to follow each step carefully, and don't hesitate to review the guide if needed. Happy solving!