Solving X²+4x-30=0 By Completing The Square A Step-by-Step Guide

Solving quadratic equations can sometimes feel like navigating a maze, with various methods available, each possessing its unique charm and applicability. One such powerful technique is completing the square, a method that not only provides solutions but also offers valuable insights into the structure of quadratic equations. In this comprehensive guide, we will embark on a step-by-step journey to solve the quadratic equation x²+4x-30=0 by masterfully employing the completing the square method. This approach is particularly useful as it lays the groundwork for understanding the quadratic formula and provides a robust method for solving equations that are not easily factorable. This exploration will enhance your problem-solving skills and deepen your comprehension of quadratic equations.

Understanding the Completing the Square Method

Before diving into the specifics of our equation, let’s first grasp the essence of completing the square. At its heart, this method is about transforming a quadratic expression into a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial. This transformation allows us to rewrite the equation in a form that is easily solvable by taking the square root. The underlying principle is to manipulate the equation algebraically until one side forms a perfect square, making the variable easily isolatable. This method is invaluable because it not only solves equations but also provides a deeper understanding of the structure and properties of quadratic expressions. The completing the square technique is a cornerstone in algebra, bridging the gap between factoring and the quadratic formula, and it's applicable to a wide range of mathematical problems, solidifying its place as an essential tool in any mathematician's arsenal. Mastering completing the square enhances your algebraic intuition and problem-solving capabilities.

Step 1: Isolate the Constant Term

The first critical step in completing the square involves isolating the constant term on one side of the equation. This means we want to move the constant term (-30 in our case) to the right side of the equation, leaving only the terms with x on the left side. To accomplish this, we add 30 to both sides of the equation x²+4x-30=0. This maintains the equation's balance while setting the stage for the next steps in the process. The modified equation now reads as x²+4x=30. This step is crucial because it sets up the quadratic expression in a form that's conducive to creating a perfect square trinomial. By isolating the constant, we prepare the quadratic and linear terms for the subsequent manipulation needed to complete the square. This seemingly simple step is a fundamental aspect of the method, as it ensures that the subsequent steps will correctly transform the equation into a solvable form. By carefully isolating the constant term, we lay a solid foundation for successfully completing the square.

Step 2: Completing the Square

Now comes the heart of the method: completing the square. This involves transforming the left side of the equation into a perfect square trinomial. To achieve this, we take half of the coefficient of our x term (which is 4), square it, and add the result to both sides of the equation. Half of 4 is 2, and 2 squared is 4. So, we add 4 to both sides of the equation x²+4x=30. This gives us x²+4x+4=30+4, which simplifies to x²+4x+4=34. The key here is that x²+4x+4 is now a perfect square trinomial, meaning it can be factored into the form (x+a)². In this case, x²+4x+4 factors to (x+2)². This transformation is the essence of the completing the square method, as it allows us to rewrite the equation in a form where we can easily isolate x. Adding the correct value to both sides not only maintains the balance of the equation but also strategically creates a perfect square, making the equation solvable. Understanding this step is crucial for mastering the technique and applying it to various quadratic equations. Completing the square in this manner is a powerful way to simplify quadratic equations and pave the way for finding solutions.

Step 3: Factor the Perfect Square Trinomial

The next step in our journey is to factor the perfect square trinomial we've created. Recall that in the previous step, we transformed x²+4x+4 into the equivalent expression (x+2)². This factorization is a critical step because it simplifies the left side of the equation, allowing us to rewrite the equation in a more manageable form. So, the equation x²+4x+4=34 now becomes (x+2)²=34. Factoring the perfect square trinomial is not just a mathematical manipulation; it's a strategic move that brings us closer to isolating the variable x. By recognizing and factoring the trinomial, we convert a complex expression into a simple squared term, which is much easier to deal with. This step underscores the elegance of the completing the square method, as it showcases how algebraic transformations can lead to significant simplifications. Mastering the ability to factor perfect square trinomials is essential for anyone seeking to solve quadratic equations effectively. Factoring the trinomial is a key to unlocking the solution.

Step 4: Take the Square Root of Both Sides

With the equation now in the form (x+2)²=34, we are ready to take the square root of both sides. This step is crucial for undoing the square and isolating the term (x+2). Remember, when we take the square root of a number, we must consider both the positive and negative roots. So, taking the square root of both sides gives us √(x+2)²=±√34, which simplifies to x+2=±√34. The introduction of both positive and negative roots is a fundamental aspect of solving equations involving squares, as it acknowledges that both values, when squared, will result in the same positive number. This step highlights the importance of paying attention to all possible solutions. Taking the square root is a pivotal point in the completing the square method, as it directly leads us to isolating x and finding the solutions to the original quadratic equation. This operation is a critical bridge between the squared term and the individual solutions for x.

Step 5: Solve for x

The final step in solving for x involves isolating x by subtracting 2 from both sides of the equation. From the previous step, we have x+2=±√34. Subtracting 2 from both sides gives us x=-2±√34. This means we have two solutions for x: x=-2+√34 and x=-2-√34. These are the exact solutions to the quadratic equation x²+4x-30=0. In this step, we've successfully isolated x, unveiling the roots of the equation. The solutions represent the points where the parabola described by the quadratic equation intersects the x-axis. By isolating x, we've not only found the answers but also completed the completing the square method, demonstrating its effectiveness in solving quadratic equations. This final step is the culmination of the entire process, showcasing the power of algebraic manipulation in revealing the solutions hidden within the equation. The values of x we've found are the definitive solutions, representing the roots of the quadratic equation.

Conclusion

In conclusion, we have successfully navigated the process of solving the quadratic equation x²+4x-30=0 by completing the square. This method, while appearing intricate at first, provides a robust and insightful way to tackle quadratic equations, especially those that don't lend themselves to simple factoring. By understanding each step – isolating the constant term, completing the square, factoring the perfect square trinomial, taking the square root of both sides, and solving for x – you can confidently apply this technique to a variety of quadratic equations. The completing the square method not only delivers solutions but also enhances your algebraic toolkit and deepens your understanding of quadratic equations. This comprehensive guide has equipped you with the knowledge and confidence to tackle similar problems, solidifying your grasp of this powerful mathematical technique. Remember, practice makes perfect, so continue to apply this method to various quadratic equations to hone your skills and mastery.