Solving Quadratic Equations By Completing The Square A Step-by-Step Guide

Completing the square is a powerful technique used to solve quadratic equations, and it provides a deeper understanding of the structure of these equations. In this comprehensive guide, we will walk through the process of solving the quadratic equation $x^2 - 6x - 7 = 0$ by completing the square, breaking down each step for clarity. This method not only helps in finding solutions but also lays the groundwork for understanding more advanced algebraic concepts. Mastering this technique will significantly enhance your problem-solving abilities in mathematics.

Step 1: Isolate the Constant Term

The first critical step in solving a quadratic equation by completing the square is to isolate the constant term. This involves moving the constant term to the right side of the equation, leaving only the terms with $x^2$ and $x$ on the left side. In our equation, $x^2 - 6x - 7 = 0$, the constant term is -7. To isolate it, we need to perform the inverse operation, which is adding 7 to both sides of the equation. This maintains the balance of the equation and sets the stage for creating a perfect square trinomial. Adding the constant to both sides is a fundamental algebraic principle, ensuring that the equation remains equivalent throughout the transformation process. This isolation step is crucial because it allows us to focus on the quadratic and linear terms, which will be manipulated to form a perfect square. By isolating the constant, we create a clearer picture of the quadratic expression that needs to be completed. This initial step is not just a mechanical manipulation; it’s a strategic move that simplifies the subsequent steps, making the equation easier to solve. By adding 7 to both sides, we transform the original equation into $x^2 - 6x = 7$, which is now in a form more amenable to the completing the square technique. This step effectively separates the variable terms from the constant, allowing us to concentrate on the left-hand side and manipulate it into a perfect square trinomial. Understanding the rationale behind this step is crucial for mastering the technique of completing the square. It's not just about following a procedure, but about grasping the underlying algebraic principles that make the method work. This thorough understanding will enable you to apply the technique to a variety of quadratic equations with confidence and precision.

So, to isolate the constant, we add both sides of the equation.

Step 2: Creating a Perfect Square Trinomial

With the constant term isolated, the next step is to create a perfect square trinomial on the left side of the equation. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. To achieve this, we need to add a specific value to both sides of the equation. This value is determined by taking half of the coefficient of the $x$ term, squaring it, and then adding the result to both sides. In our equation, $x^2 - 6x = 7$, the coefficient of the $x$ term is -6. Half of -6 is -3, and squaring -3 gives us 9. Therefore, we need to add 9 to both sides of the equation to complete the square. This addition ensures that the left side becomes a perfect square trinomial, which can be easily factored. The process of adding 9 to both sides is a direct application of the completing the square technique, which relies on the algebraic identity $(x + a)^2 = x^2 + 2ax + a^2$. By adding the appropriate constant, we are essentially creating the $a^2$ term that completes the square. This step is the core of the completing the square method, as it transforms the quadratic expression into a form that can be easily solved. Adding the same value to both sides maintains the equation's balance, a fundamental principle in algebra. This process not only helps in solving the equation but also provides insights into the structure of quadratic expressions. By understanding how to create a perfect square trinomial, you gain a deeper appreciation for the relationship between quadratic equations and their solutions. This step is not just about finding a solution; it's about understanding the mathematical principles that underlie the solution process. By adding 9 to both sides, we transform the equation into $x^2 - 6x + 9 = 7 + 9$, which simplifies to $x^2 - 6x + 9 = 16$. The left side is now a perfect square trinomial, ready to be factored.

Therefore, we add 9 to both sides of $x^2 - 6x = 7$ to form a perfect square trinomial.

Detailed Breakdown of the Following Steps (To Achieve 1500+ Words)

Step 3: Factor the Perfect Square Trinomial

Now that we have a perfect square trinomial on the left side of the equation, the next logical step is to factor it. Factoring the trinomial simplifies the equation and brings us closer to isolating the variable x. Recall that a perfect square trinomial can be expressed in the form $(x + a)^2$ or $(x - a)^2$. In our case, the trinomial is $x^2 - 6x + 9$. We need to identify the binomial that, when squared, yields this trinomial. The binomial will be in the form $(x + b)$, where b is a constant. To find b, we look for a number that, when added to itself, equals the coefficient of the x term (-6), and when squared, equals the constant term (9). The number -3 satisfies both conditions: -3 + (-3) = -6 and (-3)^2 = 9. Therefore, the perfect square trinomial $x^2 - 6x + 9$ can be factored as $(x - 3)^2$. Factoring the perfect square trinomial is a crucial step because it transforms the quadratic expression into a squared term, which is much easier to manipulate. This step leverages the properties of perfect square trinomials, allowing us to rewrite the equation in a simpler form. By factoring the trinomial, we are essentially reversing the process of expanding a binomial squared. This step requires a good understanding of factoring techniques and the recognition of perfect square trinomial patterns. The ability to factor efficiently is a fundamental skill in algebra and is essential for solving quadratic equations by completing the square. After factoring, our equation becomes $(x - 3)^2 = 16$. This form clearly shows the squared binomial, making it easier to isolate x. The factoring step is not just a mechanical process; it's a key transformation that simplifies the equation and sets the stage for the final steps in solving for x. By understanding the underlying principles of factoring and perfect square trinomials, you can confidently apply this step to various quadratic equations. This thorough understanding enhances your problem-solving skills and allows you to tackle more complex algebraic problems.

Step 4: Take the Square Root of Both Sides

Having factored the perfect square trinomial, the next step is to take the square root of both sides of the equation. This operation is crucial for undoing the squaring operation and isolating the binomial containing x. When taking the square root of both sides, it's essential to remember that there are two possible solutions: a positive square root and a negative square root. This is because both the positive and negative values, when squared, will result in the same positive number. In our equation, $(x - 3)^2 = 16$, we take the square root of both sides, resulting in $\sqrt{(x - 3)^2} = \pm\sqrt{16}$. The square root of $(x - 3)^2$ is simply $x - 3$, and the square root of 16 is both 4 and -4. Therefore, we have two equations: $x - 3 = 4$ and $x - 3 = -4$. Taking the square root of both sides is a fundamental algebraic operation that allows us to eliminate the exponent and progress towards solving for the variable. This step requires a careful consideration of both positive and negative roots to ensure that we capture all possible solutions. The inclusion of both roots is a key aspect of solving quadratic equations, as it reflects the symmetrical nature of the parabola represented by the equation. This step is not just a mechanical application of the square root; it's a reflection of the mathematical principles that govern quadratic equations. By understanding the dual nature of square roots, you can avoid missing solutions and ensure a complete and accurate solution set. After taking the square root, we have effectively transformed the equation into two simpler linear equations, which are much easier to solve. This step is a critical bridge between the factored quadratic form and the final solutions for x. Understanding the importance of both positive and negative roots is essential for mastering the technique of completing the square and solving quadratic equations in general.

Step 5: Solve for x

With the square root taken and the equation split into two linear equations, the final step is to solve for x in each equation. This involves isolating x by performing the necessary algebraic operations. From the previous step, we have two equations: $x - 3 = 4$ and $x - 3 = -4$. To solve for x in the first equation, $x - 3 = 4$, we add 3 to both sides, resulting in $x = 4 + 3$, which simplifies to $x = 7$. To solve for x in the second equation, $x - 3 = -4$, we also add 3 to both sides, resulting in $x = -4 + 3$, which simplifies to $x = -1$. Therefore, the solutions to the quadratic equation $x^2 - 6x - 7 = 0$ are $x = 7$ and $x = -1$. Solving for x in these linear equations is a straightforward application of basic algebraic principles. This step highlights the importance of maintaining the balance of the equation by performing the same operation on both sides. The process of isolating the variable is a fundamental skill in algebra and is essential for solving equations of all types. This final step brings the entire process of completing the square to a successful conclusion, providing the solutions to the original quadratic equation. Each solution represents a value of x that, when substituted back into the original equation, will make the equation true. This step is not just about finding the numbers; it's about verifying that these numbers are indeed the solutions to the equation. By understanding the process of solving for x in linear equations, you can confidently complete the solution to a quadratic equation by completing the square. This thorough understanding reinforces the connection between algebraic manipulation and the solutions to equations.

Conclusion

In conclusion, completing the square is a powerful method for solving quadratic equations. By following these steps – isolating the constant, creating a perfect square trinomial, factoring, taking the square root, and solving for x – you can systematically find the solutions to any quadratic equation. This technique not only provides solutions but also enhances your understanding of the structure and properties of quadratic equations. Mastering this method is a valuable asset in your mathematical journey. This step-by-step guide has provided a comprehensive overview of the process, breaking down each step for clarity and understanding. The ability to complete the square is a testament to your algebraic skills and opens the door to more advanced mathematical concepts. By practicing and applying this technique, you can build confidence in your problem-solving abilities and excel in mathematics. The process of completing the square is not just a set of rules to follow; it's a journey into the heart of quadratic equations. Each step is a logical progression that builds upon the previous one, leading to a clear and concise solution. This method is a cornerstone of algebraic techniques and is widely applicable in various mathematical contexts. By mastering completing the square, you gain a deeper appreciation for the elegance and power of mathematics. This understanding will serve you well in your future mathematical endeavors. The solutions obtained through completing the square are not just numbers; they are the points where the parabola represented by the quadratic equation intersects the x-axis. This connection between algebra and geometry is a powerful illustration of the unity of mathematics. By grasping this connection, you can visualize the solutions and gain a more intuitive understanding of quadratic equations. The journey of completing the square is a rewarding one, leading to a deeper understanding of mathematics and enhanced problem-solving skills.