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

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The method of completing the square is a powerful technique for solving quadratic equations. It transforms a quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. In this article, we will walk through the step-by-step process of solving the quadratic equation x^2 - 6x - 7 = 0 by completing the square. Each step will be explained in detail, making the process clear and understandable. This method not only helps in finding the solutions (roots) of the equation but also provides a solid foundation for understanding more advanced mathematical concepts. Completing the square is a fundamental technique in algebra, and mastering it can significantly enhance your problem-solving skills. Whether you're a student learning algebra for the first time or someone looking to refresh your knowledge, this comprehensive guide will provide you with a clear understanding of how to complete the square.

1. Isolate the Constant Term

To begin the process of completing the square, the first step is to isolate the constant term on one side of the equation. This means moving the constant term to the right side of the equation, leaving the terms with x on the left side. In our equation, x^2 - 6x - 7 = 0, the constant term is -7. To isolate this term, we need to add 7 to both sides of the equation. Adding the same value to both sides ensures that the equation remains balanced, maintaining the equality. This is a fundamental principle in algebra, and it is crucial for solving equations correctly. By adding 7 to both sides, we eliminate the -7 on the left side, effectively moving it to the right side. This step sets the stage for the next phase of completing the square, where we will transform the left side into a perfect square trinomial. The careful execution of this initial step is essential for the subsequent steps to work correctly. Ignoring this step or performing it incorrectly can lead to errors in the final solution. Therefore, it is vital to understand why we isolate the constant and how it helps in the overall process of solving the quadratic equation. Isolating the constant is not just a mechanical step; it is a strategic move that simplifies the equation and prepares it for the core of the completing the square method.

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

2. Completing the Square

The next crucial step in solving x^2 - 6x - 7 = 0 by completing the square is to transform the left side of the equation into a perfect square trinomial. After isolating the constant term in the previous step, we now have the equation in the form x^2 - 6x = 7. To complete the square, 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 (which is -6 in our case), squaring it, and adding the result to both sides. The coefficient of the x term is -6, so half of it is -3, and squaring -3 gives us 9. This number, 9, is what we need to add to both sides of the equation to complete the square. Adding the same number to both sides keeps the equation balanced, maintaining the equality. By adding 9, we are essentially creating a perfect square trinomial on the left side of the equation. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, x^2 - 6x + 9 can be factored into (x - 3)^2. This transformation is the heart of the completing the square method. It allows us to rewrite the quadratic equation in a form that is easily solvable by taking the square root. Understanding this step is crucial for mastering the technique of completing the square. Without correctly determining and adding the appropriate value, the equation cannot be transformed into the desired perfect square trinomial form. Thus, this step requires careful attention to detail and a solid grasp of algebraic principles.

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

3. Solving the Equation

With the equation now in the form (x - 3)^2 = 16, the next step is to solve for x. To do this, we take the square root of both sides of the equation. When taking the square root, 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 square roots, when squared, will result in the same positive number. In our case, the square root of 16 is both +4 and -4. Taking the square root of (x - 3)^2 gives us (x - 3), so we have x - 3 = ±4. Now, we have two separate linear equations to solve: x - 3 = 4 and x - 3 = -4. To solve for x in each equation, we add 3 to both sides. For the first equation, x - 3 = 4, adding 3 to both sides gives us x = 7. For the second equation, x - 3 = -4, adding 3 to both sides gives us x = -1. Therefore, the solutions to the original quadratic equation x^2 - 6x - 7 = 0 are x = 7 and x = -1. These are the values of x that satisfy the equation, making it true. The final step of solving the equation involves carefully applying algebraic principles to isolate x and find its possible values. It highlights the importance of considering both positive and negative roots when dealing with square roots in equations.

Conclusion

In conclusion, solving the quadratic equation x^2 - 6x - 7 = 0 by completing the square involves a series of well-defined steps. First, we isolate the constant term by adding 7 to both sides of the equation. Second, we complete the square by adding 9 to both sides, which transforms the left side into a perfect square trinomial. Finally, we solve for x by taking the square root of both sides and considering both positive and negative roots. This method is not just a way to find solutions; it also provides a deeper understanding of the structure of quadratic equations and their properties. Mastering the technique of completing the square is a valuable skill in algebra, and it forms a foundation for more advanced mathematical concepts. By following these steps carefully and understanding the underlying principles, you can confidently solve a wide range of quadratic equations. The process emphasizes the importance of algebraic manipulation, maintaining equality, and considering all possible solutions. Completing the square is a testament to the elegance and power of algebraic methods in problem-solving.