Solving $7x - 9 = 7x^2 - 49x$ By Completing The Square A Step-by-Step Guide
In the realm of mathematics, quadratic equations hold a significant position, appearing in various fields from physics to engineering. One powerful technique for solving these equations is completing the square. This method not only provides solutions but also offers valuable insights into the structure and properties of quadratic functions. In this comprehensive guide, we will delve into the step-by-step process of solving the quadratic equation $7x - 9 = 7x^2 - 49x$ by completing the square. This method involves transforming the equation into a perfect square trinomial, which can then be easily solved. Let's embark on this mathematical journey and unlock the secrets of completing the square.
Understanding the Quadratic Equation
Before diving into the solution, let's first understand the given quadratic equation: $7x - 9 = 7x^2 - 49x$. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form $ax^2 + bx + c = 0$, where a, b, and c are constants and a ≠0. To effectively solve this equation by completing the square, we need to rearrange it into the standard quadratic form. This involves moving all terms to one side of the equation, leaving zero on the other side. By doing so, we set the stage for the subsequent steps in the method. This initial rearrangement is crucial as it allows us to identify the coefficients and constant terms necessary for completing the square. The standard form provides a clear framework for manipulating the equation and ultimately finding its solutions.
Rearranging the Equation
Our initial step involves rearranging the given equation $7x - 9 = 7x^2 - 49x$ into the standard quadratic form. To achieve this, we subtract $7x$ from both sides and add 9 to both sides of the equation. This process effectively moves all terms to the right-hand side, leaving zero on the left-hand side. The result is a quadratic equation in the form $0 = 7x^2 - 56x + 9$. This form is essential because it aligns with the general structure $ax^2 + bx + c = 0$, which is the foundation for applying the method of completing the square. By carefully rearranging the equation, we set the stage for the next steps, ensuring that the subsequent manipulations are mathematically sound and lead us closer to the solution. This transformation is a fundamental prerequisite for successfully completing the square.
Completing the Square: A Step-by-Step Approach
Now that we have the equation in the standard form, we can proceed with the core process of completing the square. This technique involves transforming the quadratic expression into a perfect square trinomial, which is an expression that can be factored into the square of a binomial. The key idea behind completing the square is to manipulate the equation algebraically until one side becomes a perfect square, making it easier to solve for the variable. This method is particularly useful when the quadratic equation cannot be easily factored using traditional techniques. By understanding the underlying principles of perfect square trinomials, we can systematically apply this method to a wide range of quadratic equations. The following steps will guide us through the process of completing the square for our specific equation.
Step 1: Isolate the Quadratic and Linear Terms
The first step in completing the square is to isolate the quadratic and linear terms on one side of the equation. In our rearranged equation, $0 = 7x^2 - 56x + 9$, we want to isolate the terms $7x^2$ and $-56x$. To do this, we subtract 9 from both sides of the equation, resulting in $-9 = 7x^2 - 56x$. This step is crucial because it prepares the equation for the subsequent steps where we will manipulate the expression to form a perfect square trinomial. By isolating the quadratic and linear terms, we create a clear focus on the part of the equation that needs to be transformed. This separation is a fundamental aspect of the method of completing the square and ensures that the following algebraic manipulations are targeted and effective.
Step 2: Factor out the Leading Coefficient
Next, we need to factor out the leading coefficient from the quadratic and linear terms. In our equation, $-9 = 7x^2 - 56x$, the leading coefficient is 7. We factor out 7 from the terms on the right-hand side, which gives us $-9 = 7(x^2 - 8x)$. This step is essential because completing the square requires the coefficient of the $x^2$ term to be 1. By factoring out the leading coefficient, we ensure that the expression inside the parentheses is in the correct form for completing the square. This manipulation simplifies the subsequent calculations and makes the process more manageable. Factoring out the leading coefficient is a critical step in preparing the quadratic expression for the transformation into a perfect square trinomial.
Step 3: Complete the Square
Now comes the core of the method: completing the square. We focus on the expression inside the parentheses, which is $x^2 - 8x$. To complete the square, we need to add a constant term that will make this expression a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. To find the constant term, we take half of the coefficient of the x term, which is -8, and square it. Half of -8 is -4, and squaring -4 gives us 16. Thus, we need to add 16 inside the parentheses to complete the square. However, since we are adding 16 inside the parentheses, which are multiplied by 7, we are effectively adding 7 * 16 = 112 to the right-hand side of the equation. Therefore, we must also add 112 to the left-hand side to maintain the balance of the equation. This gives us $-9 + 112 = 7(x^2 - 8x + 16)$. This step is the heart of the method, where we strategically add a constant to both sides of the equation to create a perfect square trinomial, making the equation solvable.
Step 4: Factor the Perfect Square Trinomial
After completing the square, we now have a perfect square trinomial on the right-hand side of the equation. Our equation is $-9 + 112 = 7(x^2 - 8x + 16)$. The expression $x^2 - 8x + 16$ is a perfect square trinomial because it can be factored into $(x - 4)^2$. So, we rewrite the equation as $103 = 7(x - 4)^2$. This step simplifies the equation significantly, as we have transformed the quadratic expression into a squared term. Factoring the perfect square trinomial is a crucial step in the process, as it sets the stage for isolating the variable and finding the solutions. This transformation is a direct result of completing the square and allows us to proceed with solving the equation in a straightforward manner.
Step 5: Solve for x
Now that we have factored the perfect square trinomial, we can solve for x. Our equation is $103 = 7(x - 4)^2$. First, we divide both sides by 7 to isolate the squared term, which gives us $ rac103}{7} = (x - 4)^2$. Next, we take the square root of both sides, remembering to consider both the positive and negative square roots7}} = x - 4$. Finally, we add 4 to both sides to isolate x{7}}$. These are the solutions to the quadratic equation. Solving for x involves carefully undoing the operations performed on the variable, step by step. By isolating the squared term, taking the square root, and then adding the constant, we arrive at the solutions, which represent the values of x that satisfy the original quadratic equation.
Final Solutions
Therefore, the solutions to the quadratic equation $7x - 9 = 7x^2 - 49x$ obtained by completing the square are $x = 4 + \sqrt{\frac{103}{7}}$ and $x = 4 - \sqrt{\frac{103}{7}}$. These values represent the points where the parabola represented by the quadratic equation intersects the x-axis. Completing the square is a powerful method that not only provides the solutions but also offers a deeper understanding of the structure of quadratic equations. By transforming the equation into a perfect square trinomial, we can easily solve for the variable and gain insights into the behavior of the quadratic function. The solutions we have found are the roots of the equation, and they are essential in various mathematical and real-world applications.
In conclusion, solving quadratic equations by completing the square is a systematic and effective method that allows us to find the solutions even when traditional factoring is not straightforward. By following the steps outlined in this guide, you can confidently tackle any quadratic equation and unlock its solutions. This technique is a valuable tool in the mathematician's arsenal and provides a solid foundation for further exploration of quadratic functions and their applications.
