Solving 7x - 9 = 7x² - 49x By Completing The Square A Step-by-Step Guide
Introduction to Quadratic Equations and Completing the Square
In the realm of mathematics, quadratic equations hold a prominent position due to their wide applicability in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term that is squared. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Solving a quadratic equation involves finding the values of the variable x that satisfy the equation, also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method has its own advantages and is suitable for different types of quadratic equations. This article will delve into the method of completing the square, which is particularly useful for transforming a quadratic equation into a form where the solutions can be easily determined. This method not only provides the solutions but also helps in understanding the structure and properties of quadratic equations. The process of completing the square involves manipulating the quadratic equation algebraically to create a perfect square trinomial on one side of the equation. This transformation allows us to rewrite the equation in a form that can be easily solved by taking the square root. The technique is fundamental in algebra and provides a strong foundation for more advanced mathematical concepts. Understanding completing the square enhances problem-solving skills and offers a deeper insight into the nature of quadratic equations. The steps involved in completing the square are systematic and can be applied to any quadratic equation, making it a versatile tool in a mathematician's arsenal. This article will guide you through these steps in detail, providing clear explanations and examples to ensure a thorough understanding of the method. By mastering completing the square, you will be able to tackle a wide range of quadratic equations with confidence and precision. Furthermore, this method is not just limited to solving equations; it also has applications in graphing parabolas and understanding the vertex form of a quadratic function.
Problem Statement: 7x - 9 = 7x² - 49x
Our specific task is to solve the quadratic equation 7x - 9 = 7x² - 49x using the method of completing the square. This equation presents a classic example of a quadratic equation that can be solved effectively by this method. The first step in tackling any quadratic equation is to rearrange it into the standard form, which is ax² + bx + c = 0. This standard form allows us to easily identify the coefficients a, b, and c, which are crucial for applying various solution methods, including completing the square. To transform the given equation into the standard form, we need to move all terms to one side of the equation, leaving zero on the other side. This involves algebraic manipulation, such as adding or subtracting terms from both sides, to ensure that the equation remains balanced. Once the equation is in the standard form, we can proceed with the method of completing the square. This method involves creating a perfect square trinomial on one side of the equation, which simplifies the process of finding the solutions. The challenge with the equation 7x - 9 = 7x² - 49x lies in the presence of both linear and quadratic terms, which necessitates a systematic approach to isolate the variable x. By following the steps of completing the square, we can transform this equation into a more manageable form, ultimately leading to the solutions. The coefficients in this particular equation play a significant role in determining the complexity of the steps involved in completing the square. The coefficient of the x² term, a, needs to be considered when making the leading coefficient 1, which is a crucial step in the process. Additionally, the coefficients b and c influence the values that need to be added and subtracted to complete the square. Understanding the interplay of these coefficients is key to successfully applying the method. This problem provides an excellent opportunity to demonstrate the power and elegance of completing the square as a method for solving quadratic equations. By working through this example, we can gain a deeper appreciation for the algebraic techniques involved and their practical application in solving mathematical problems. The subsequent sections will provide a detailed, step-by-step solution to the equation, ensuring clarity and understanding of each stage of the process.
Step 1: Rearrange the Equation into Standard Form
The initial step in solving the quadratic equation 7x - 9 = 7x² - 49x is to rearrange it into the standard form ax² + bx + c = 0. This form is essential because it allows us to clearly identify the coefficients a, b, and c, which are necessary for applying the method of completing the square. To achieve this, we need to move all the terms to one side of the equation, leaving zero on the other side. This involves performing algebraic operations on both sides of the equation to maintain the balance. Starting with the given equation, 7x - 9 = 7x² - 49x, we want to have all terms on the right-hand side. To do this, we subtract 7x from both sides of the equation. This gives us -9 = 7x² - 49x - 7x. Next, we simplify the right-hand side by combining the like terms, which are the terms with x. The equation now becomes -9 = 7x² - 56x. To complete the rearrangement, we add 9 to both sides of the equation. This results in 0 = 7x² - 56x + 9. Now, the equation is in the standard form ax² + bx + c = 0, where a = 7, b = -56, and c = 9. It's often a good practice to rewrite the equation with the zero on the right-hand side for clarity, so we can write the equation as 7x² - 56x + 9 = 0. This rearranged equation is now ready for the next steps in the method of completing the square. The process of rearranging the equation is crucial because it sets the stage for the subsequent steps. Without the equation in the standard form, it would be difficult to apply the completing the square method effectively. This step ensures that we have a clear and organized starting point for solving the quadratic equation. The standard form also helps in visualizing the structure of the equation and identifying the key components needed for further calculations. By carefully following this step, we lay a solid foundation for the rest of the solution process. The correct identification of the coefficients a, b, and c is vital for the accuracy of the subsequent steps, making this rearrangement a critical part of the overall solution.
Step 2: Divide by the Leading Coefficient (if necessary)
After rearranging the quadratic equation into the standard form ax² + bx + c = 0, the next crucial step in completing the square is to ensure that the leading coefficient, a, is equal to 1. This is necessary because the method of completing the square relies on having a coefficient of 1 for the x² term. In our equation, 7x² - 56x + 9 = 0, the leading coefficient a is 7, which is not equal to 1. Therefore, we need to divide the entire equation by 7 to make the leading coefficient 1. Dividing each term of the equation by 7, we get: (7x²)/7 - (56x)/7 + 9/7 = 0/7. This simplifies to x² - 8x + 9/7 = 0. Now, the equation has a leading coefficient of 1, and we can proceed with the next steps of completing the square. This step is essential because it simplifies the process of creating a perfect square trinomial. When the leading coefficient is 1, we can directly manipulate the x term and the constant term to form a perfect square. If the leading coefficient were not 1, the process would be more complex and involve additional steps to account for the coefficient. Dividing by the leading coefficient ensures that the subsequent steps are straightforward and less prone to errors. It's important to note that this step is only necessary if the leading coefficient is not already 1. If the equation started with a leading coefficient of 1, this step would be skipped, and we would move directly to the next step. However, in this case, dividing by 7 is crucial for making the completing the square method applicable. The resulting equation, x² - 8x + 9/7 = 0, is now in a form that is much easier to work with. The absence of a coefficient in front of the x² term simplifies the calculations involved in finding the value that needs to be added and subtracted to complete the square. This step is a fundamental part of the completing the square method and ensures that the rest of the solution process is as efficient and accurate as possible. By correctly performing this division, we set the stage for creating a perfect square trinomial and ultimately solving the quadratic equation.
Step 3: Move the Constant Term to the Right Side
With the quadratic equation now in the form x² - 8x + 9/7 = 0, where the leading coefficient is 1, the next step in completing the square is to isolate the terms containing x on one side of the equation and move the constant term to the other side. This is done to prepare the equation for the creation of a perfect square trinomial. In our equation, the constant term is 9/7. To move it to the right side of the equation, we subtract 9/7 from both sides. This gives us: x² - 8x + 9/7 - 9/7 = 0 - 9/7. Simplifying this, we get x² - 8x = -9/7. Now, we have the x² and x terms on the left side, and the constant term on the right side. This arrangement is crucial for the next step, where we will add a specific value to both sides to complete the square. Moving the constant term to the right side of the equation is a strategic step that allows us to focus on the left side and manipulate it into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)². By isolating the x terms, we can determine what value needs to be added to both sides to create this perfect square. The constant term on the right side will then be adjusted accordingly. This step is relatively straightforward but is essential for the success of the completing the square method. It simplifies the subsequent calculations and allows us to clearly see the structure of the equation as we progress towards the solution. By correctly moving the constant term, we create a clear separation between the terms we need to manipulate and the constant value that will result from completing the square. This separation makes the process more organized and easier to follow. The equation x² - 8x = -9/7 is now in a form that is perfectly suited for the next step, which involves finding the value needed to complete the square and adding it to both sides of the equation. This step is a critical part of the overall method and sets the stage for finding the solutions to the quadratic equation.
Step 4: Complete the Square
After moving the constant term to the right side, our quadratic equation is now in the form x² - 8x = -9/7. The critical step of completing the square involves adding a specific value to both sides of the equation to create a perfect square trinomial on the left side. A perfect square trinomial is a trinomial that can be factored into the form (x + a)² or (x - a)². To find the value needed to complete the square, we take half of the coefficient of the x term, square it, and add the result to both sides of the equation. In our equation, the coefficient of the x term is -8. Half of -8 is -4, and squaring -4 gives us (-4)² = 16. Therefore, we need to add 16 to both sides of the equation to complete the square. Adding 16 to both sides, we get: x² - 8x + 16 = -9/7 + 16. Now, the left side of the equation, x² - 8x + 16, is a perfect square trinomial. It can be factored as (x - 4)². On the right side of the equation, we need to add -9/7 and 16. To do this, we need a common denominator. We can rewrite 16 as 16/1, and then multiply the numerator and denominator by 7 to get 112/7. So, -9/7 + 16 = -9/7 + 112/7 = 103/7. Thus, our equation becomes (x - 4)² = 103/7. This equation is now in a form that is easily solvable by taking the square root of both sides. Completing the square is a fundamental technique in algebra and is crucial for solving quadratic equations. This step transforms the equation into a form where the variable x is part of a perfect square, making it possible to isolate x and find the solutions. The value that we add to both sides, in this case, 16, is carefully chosen to ensure that the left side becomes a perfect square trinomial. This process may seem abstract at first, but it is a powerful tool for manipulating quadratic equations and finding their solutions. By correctly completing the square, we have simplified the equation and set the stage for the final steps of solving for x. The equation (x - 4)² = 103/7 is now in a form that is much easier to work with, and the subsequent steps will involve taking the square root and isolating x to find the solutions to the original quadratic equation.
Step 5: Take the Square Root of Both Sides
Having completed the square, our equation is now in the form (x - 4)² = 103/7. The next step in solving for x is to take the square root of both sides of the equation. This will help us eliminate the square on the left side and isolate the term (x - 4). When taking the square root of both sides, it's crucial to remember that we need to consider both the positive and negative square roots. This is because both the positive and negative values, when squared, will give us the same result. Taking the square root of both sides, we get: √(x - 4)² = ±√(103/7). This simplifies to x - 4 = ±√(103/7). Now, we have isolated the term (x - 4) on the left side, but we still have a square root on the right side. To simplify the square root, we can rewrite √(103/7) as √103 / √7. To rationalize the denominator, we multiply both the numerator and denominator by √7, which gives us (√103 * √7) / (√7 * √7) = √(103 * 7) / 7 = √721 / 7. So, our equation becomes x - 4 = ±(√721 / 7). Taking the square root of both sides is a critical step in solving quadratic equations by completing the square. It allows us to remove the square from the perfect square trinomial and move closer to isolating x. Remembering to include both the positive and negative square roots is essential because it ensures that we find all possible solutions to the equation. The square root operation can sometimes introduce irrational numbers, as seen in our example with √103 and √7. Rationalizing the denominator, as we did by multiplying by √7, is a common practice to simplify the expression and make it easier to work with. The equation x - 4 = ±(√721 / 7) is now in a form that is very close to the final solution. The only remaining step is to isolate x by adding 4 to both sides of the equation. This will give us the two possible values of x that satisfy the original quadratic equation. By carefully taking the square root and simplifying the resulting expressions, we have made significant progress towards finding the solutions.
Step 6: Solve for x
After taking the square root of both sides, our equation is now x - 4 = ±(√721 / 7). The final step in solving for x is to isolate x by adding 4 to both sides of the equation. This will give us the two possible values of x that satisfy the original quadratic equation. Adding 4 to both sides, we get: x = 4 ± (√721 / 7). This equation represents two solutions for x: one where we add (√721 / 7) to 4, and another where we subtract (√721 / 7) from 4. To express these solutions separately, we can write: x₁ = 4 + (√721 / 7) and x₂ = 4 - (√721 / 7). These are the exact solutions to the quadratic equation. If we need approximate solutions, we can use a calculator to find the approximate value of √721, which is approximately 26.85. Then, we can calculate the approximate values of x₁ and x₂: x₁ ≈ 4 + (26.85 / 7) ≈ 4 + 3.84 ≈ 7.84 x₂ ≈ 4 - (26.85 / 7) ≈ 4 - 3.84 ≈ 0.16 Therefore, the approximate solutions for x are 7.84 and 0.16. Solving for x is the culmination of the completing the square method. It involves isolating the variable to find the values that satisfy the original equation. The step of adding 4 to both sides is a straightforward algebraic manipulation, but it is crucial for obtaining the final solutions. The presence of the ± sign indicates that there are two possible solutions, which is typical for quadratic equations. These solutions can be expressed in exact form, as we did with 4 ± (√721 / 7), or in approximate form, as we calculated using a calculator. Understanding how to express solutions in both forms is important for different applications and contexts. The exact solutions are precise and do not involve any rounding, while the approximate solutions are useful for practical purposes where a numerical value is needed. By successfully solving for x, we have completed the process of solving the quadratic equation 7x - 9 = 7x² - 49x using the method of completing the square. This method provides a systematic way to find the solutions to quadratic equations and is a valuable tool in algebra.
Conclusion
In conclusion, we have successfully solved the quadratic equation 7x - 9 = 7x² - 49x using the method of completing the square. This method involves a series of algebraic manipulations to transform the equation into a form where the solutions can be easily determined. The steps we followed were: rearranging the equation into the standard form ax² + bx + c = 0, dividing by the leading coefficient if necessary to make it equal to 1, moving the constant term to the right side of the equation, completing the square by adding the square of half the coefficient of the x term to both sides, taking the square root of both sides, and finally, solving for x to find the two possible solutions. The exact solutions we found were x = 4 ± (√721 / 7), and the approximate solutions were x ≈ 7.84 and x ≈ 0.16. Completing the square is a powerful technique for solving quadratic equations because it provides a systematic approach that can be applied to any quadratic equation, regardless of whether it can be easily factored. This method is particularly useful when the quadratic equation does not factor neatly, as it allows us to find the solutions even when they are irrational or complex numbers. Furthermore, completing the square is not just a method for solving equations; it also has applications in other areas of mathematics, such as graphing parabolas and finding the vertex of a quadratic function. The process of completing the square helps to deepen our understanding of quadratic equations and their properties. By transforming the equation into a perfect square trinomial, we gain insight into the structure of the equation and the relationship between its coefficients and solutions. This method also reinforces important algebraic skills, such as manipulating equations, working with fractions and square roots, and applying the principles of equality. Mastering the method of completing the square is an essential skill for anyone studying algebra and beyond. It provides a solid foundation for more advanced mathematical concepts and enhances problem-solving abilities. By understanding and practicing this method, you can confidently tackle a wide range of quadratic equations and gain a deeper appreciation for the beauty and power of mathematics. The detailed step-by-step solution provided in this article serves as a comprehensive guide for solving quadratic equations by completing the square, and it can be used as a reference for future problems.
