Solving X^2-6x-7=0 By Completing The Square A Step-by-Step Guide

This comprehensive guide will walk you through the process of solving the quadratic equation $x^2 - 6x - 7 = 0$ by completing the square. Completing the square is a powerful technique for solving quadratic equations, and it provides a strong foundation for understanding the quadratic formula. This method is especially useful when the quadratic equation is not easily factorable. We will break down each step in detail, ensuring you understand the logic and mechanics behind the process. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide a clear and thorough explanation.

1. Understanding Quadratic Equations and Completing the Square

Before we dive into the specifics of solving $x^2 - 6x - 7 = 0$, let's understand the basics. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a eq 0$. The solutions to the quadratic equation are also known as the roots or zeros of the equation. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Completing the square is a particularly insightful method because it allows us to rewrite the quadratic equation in a form that makes it easy to find the solutions. The main idea behind completing the square is to manipulate the quadratic expression into a perfect square trinomial, which can then be factored into the form $(x + k)^2$ or $(x - k)^2$. This transformation allows us to isolate the variable $x$ and solve for its values. This method not only helps in finding the solutions but also provides a deeper understanding of the structure of quadratic equations. It is a fundamental technique in algebra and has applications in various fields of mathematics and beyond. By mastering completing the square, you gain a versatile tool for solving a wide range of problems and a stronger understanding of algebraic manipulations. The process involves several steps, each building on the previous one, to transform the equation into a solvable form. Understanding each step thoroughly is essential for successfully applying this method.

2. Step 1 Isolate the Constant Term

The initial step in completing the square involves isolating the constant term. In the given equation, $x^2 - 6x - 7 = 0$, the constant term is -7. To isolate this constant, we need to eliminate it from the left side of the equation and move it to the right side. This is achieved by performing the inverse operation of subtraction, which is addition. Specifically, we add 7 to both sides of the equation. This maintains the balance of the equation, ensuring that we are performing a valid algebraic manipulation. Adding 7 to both sides of $x^2 - 6x - 7 = 0$ gives us: $x^2 - 6x - 7 + 7 = 0 + 7$. Simplifying this, we get $x^2 - 6x = 7$. Now, the constant term is isolated on the right side of the equation, and we have a quadratic expression on the left side. This sets the stage for the next step, where we will work on completing the square on the left side. It’s crucial to perform this step accurately, as any error here will propagate through the rest of the solution. Isolating the constant term is a fundamental step because it allows us to focus on the quadratic and linear terms and transform them into a perfect square trinomial. Without this initial isolation, the subsequent steps would be significantly more complex. This process ensures that we are working with the core components needed to complete the square effectively. The result of this step, $x^2 - 6x = 7$, is the foundation upon which we build the perfect square.

3. Step 2 Completing the Square by Adding $(\frac{b}{2})^2$

Now that we have isolated the constant term, the next crucial step is to complete the square. This involves transforming the left side of the equation, $x^2 - 6x$, into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + k)^2$ or $(x - k)^2$. 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 (the $b$ term), squaring it, and adding the result to both sides. In our equation, $x^2 - 6x = 7$, the coefficient of the $x$ term is -6. So, we take half of -6, which is -3, and then square it: $(-3)^2 = 9$. This means we need to add 9 to both sides of the equation to complete the square. Adding 9 to both sides gives us: $x^2 - 6x + 9 = 7 + 9$. On the left side, we now have the perfect square trinomial $x^2 - 6x + 9$, and on the right side, we have $7 + 9 = 16$. This results in the equation $x^2 - 6x + 9 = 16$. The left side can be factored into $(x - 3)^2$, which demonstrates that we have successfully completed the square. The right side is simply a constant, making the equation easier to solve. Completing the square is a strategic manipulation that allows us to rewrite the quadratic equation in a more manageable form. By adding the correct value, we transform the left side into a perfect square, which is a key step in isolating $x$ and finding the solutions.

4. Step 3 Factoring the Perfect Square Trinomial

After completing the square, we now have the equation $x^2 - 6x + 9 = 16$. The next step is to factor the perfect square trinomial on the left side. By definition, a perfect square trinomial can be factored into the form $(x + k)^2$ or $(x - k)^2$. In our case, the trinomial $x^2 - 6x + 9$ fits this pattern. We need to find a value that, when squared, gives us 9 and, when doubled, gives us -6. This value is -3. Therefore, we can factor the trinomial as $(x - 3)^2$. So, our equation becomes $(x - 3)^2 = 16$. Factoring the perfect square trinomial is a critical step because it simplifies the equation into a form where we can easily isolate $x$. By rewriting the left side as a squared term, we prepare the equation for the next step, which involves taking the square root of both sides. This simplification is the core advantage of completing the square. It transforms a complex quadratic expression into a simpler, more manageable form. Accurate factoring is essential here; any mistake in factoring will lead to incorrect solutions. The ability to recognize and factor perfect square trinomials is a fundamental skill in algebra, and this step highlights its importance in solving quadratic equations. By factoring the trinomial correctly, we set the stage for the final steps in solving for $x$, bringing us closer to the roots of the equation.

5. Step 4 Taking the Square Root of Both Sides

Now that we have the equation $(x - 3)^2 = 16$, the next step is to take the square root of both sides. This operation will help us eliminate the square on the left side and move closer to isolating $x$. When we take the square root of a squared term, we must consider both the positive and negative roots, since both positive and negative values, when squared, will result in a positive number. Therefore, taking the square root of both sides of $(x - 3)^2 = 16$ gives us: $\sqrt{(x - 3)^2} = \pm\sqrt{16}$. Simplifying this, we get $x - 3 = \pm 4$. It’s crucial to remember the $\pm$ sign, as this represents the two possible solutions for $x$. Forgetting this step will result in missing one of the roots of the equation. Taking the square root of both sides is a key step in undoing the square operation and revealing the linear expression inside the parentheses. This step bridges the gap between the squared form and the final solution. By considering both positive and negative roots, we ensure that we capture all possible values of $x$ that satisfy the original equation. This step is a direct application of the properties of square roots and is essential for correctly solving equations involving squared terms. The resulting equation, $x - 3 = \pm 4$, sets up the final step of isolating $x$ and finding the two solutions.

6. Step 5 Solving for x

After taking the square root of both sides, we have the equation $x - 3 = \pm 4$. The final step is to solve for x. This involves isolating $x$ by performing the necessary algebraic operations. We have two cases to consider due to the $\pm$ sign: Case 1: $x - 3 = 4$ Case 2: $x - 3 = -4$ For Case 1, we add 3 to both sides of the equation $x - 3 = 4$ to isolate $x$: $x - 3 + 3 = 4 + 3$ This simplifies to $x = 7$. For Case 2, we add 3 to both sides of the equation $x - 3 = -4$ to isolate $x$: $x - 3 + 3 = -4 + 3$ This simplifies to $x = -1$. Therefore, the two solutions for the equation $x^2 - 6x - 7 = 0$ are $x = 7$ and $x = -1$. These are the values of $x$ that satisfy the original equation. Solving for $x$ is the culmination of all the previous steps. By carefully considering both positive and negative roots, we arrive at the complete set of solutions. This step demonstrates the power of algebraic manipulation in isolating variables and finding the values that make the equation true. Each solution represents a point where the quadratic function intersects the x-axis. Verifying these solutions by substituting them back into the original equation can confirm their correctness. The final answers, $x = 7$ and $x = -1$, provide a complete solution to the quadratic equation using the method of completing the square.

7. Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 - 6x - 7 = 0$ by completing the square. This method involves several key steps: isolating the constant term, completing the square by adding $(\frac{b}{2})^2$ to both sides, factoring the perfect square trinomial, taking the square root of both sides, and solving for $x$. By following these steps carefully, we found the two solutions to be $x = 7$ and $x = -1$. Completing the square is a powerful technique for solving quadratic equations, especially when factoring is not straightforward. It provides a clear and systematic approach to finding the roots of a quadratic equation and lays a strong foundation for understanding more advanced algebraic concepts. The method not only helps in finding the solutions but also enhances the understanding of the structure and properties of quadratic equations. Mastering completing the square is a valuable skill for anyone studying algebra and mathematics in general. It provides a versatile tool for solving a wide range of problems and a deeper insight into algebraic manipulations. By understanding each step and practicing the technique, you can confidently solve quadratic equations using this method. The solutions obtained, $x = 7$ and $x = -1$, are the points where the quadratic function intersects the x-axis, providing a complete picture of the equation's behavior. This comprehensive guide has walked you through each step, ensuring you have a solid understanding of how to solve quadratic equations by completing the square.