Solve 4x^2 + 16x - 84 = 0 By Completing The Square

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Solving quadratic equations is a fundamental skill in algebra, and completing the square is a powerful method to achieve this. In this article, we will demonstrate how to solve the quadratic equation 4x2+16xβˆ’84=04x^2 + 16x - 84 = 0 by completing the square. This method not only helps find the solutions but also provides insight into the structure of quadratic equations. Let’s dive into the step-by-step process to solve this equation and understand the underlying concepts.

Understanding the Method of Completing the Square

Before we tackle the specific equation, it’s crucial to understand the general method of completing the square. Completing the square is a technique used to rewrite a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0 into the form a(x+h)2+k=0a(x + h)^2 + k = 0, where hh and kk are constants. This form allows us to easily solve for xx by isolating the squared term and taking the square root. The method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a binomial squared. This approach is particularly useful when the quadratic equation cannot be easily factored using other methods. By understanding this method, we can transform complex quadratic equations into a more manageable form, enabling us to find the solutions efficiently and accurately. This method not only helps in solving equations but also in understanding the properties of quadratic functions and their graphs.

The core idea behind completing the square is to manipulate the given quadratic expression to form a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, x2+2x+1x^2 + 2x + 1 is a perfect square trinomial because it can be factored into (x+1)2(x + 1)^2. The process involves adding and subtracting a specific constant term to the quadratic expression to create this perfect square trinomial. This constant is determined by taking half of the coefficient of the xx term and squaring it. This step is crucial because it transforms the original quadratic expression into a form that can be easily factored and solved. By understanding how to manipulate quadratic expressions in this way, we gain a powerful tool for solving a wide range of quadratic equations.

Completing the square is more than just a method for solving equations; it is a fundamental technique that provides insights into the structure and properties of quadratic functions. It helps in converting the standard form of a quadratic equation, ax2+bx+c=0ax^2 + bx + c = 0, into the vertex form, a(xβˆ’h)2+k=0a(x - h)^2 + k = 0, where (h,k)(h, k) represents the vertex of the parabola. This vertex form is incredibly useful for graphing quadratic functions and determining their maximum or minimum values. By completing the square, we can easily identify the vertex and axis of symmetry of the parabola, which are key features for understanding the function's behavior. Furthermore, completing the square is a foundational concept for deriving the quadratic formula, which is a general solution for any quadratic equation. This connection highlights the importance of mastering this technique for a comprehensive understanding of quadratic functions and their applications in various fields, such as physics, engineering, and economics.

Step-by-Step Solution of 4x^2 + 16x - 84 = 0

Let's now apply the method of completing the square to solve the equation 4x2+16xβˆ’84=04x^2 + 16x - 84 = 0. We will break down the solution into clear, manageable steps to ensure a thorough understanding of the process. Each step is crucial in transforming the equation into a solvable form, and by carefully following these steps, you can confidently tackle similar quadratic equations. The process involves factoring, adding and subtracting constants, and ultimately rewriting the equation in a form where the solutions can be easily determined. This step-by-step approach not only helps in finding the correct answers but also reinforces the underlying principles of completing the square.

Step 1: Divide by the Leading Coefficient

The first step in completing the square is to ensure that the coefficient of the x2x^2 term is 1. In our equation, 4x2+16xβˆ’84=04x^2 + 16x - 84 = 0, the leading coefficient is 4. To make it 1, we divide the entire equation by 4:

rac{4x^2}{4} + rac{16x}{4} - rac{84}{4} = rac{0}{4}

This simplifies to:

x2+4xβˆ’21=0x^2 + 4x - 21 = 0

This step is crucial because the subsequent steps of completing the square are designed to work with a quadratic equation where the leading coefficient is 1. By dividing by the leading coefficient, we simplify the equation and make it easier to manipulate. This initial simplification sets the stage for the remaining steps and ensures that the process of completing the square can be applied effectively. It also helps in avoiding unnecessary complications that can arise when working with larger coefficients.

Step 2: Move the Constant Term to the Right Side

Next, we need to isolate the terms containing xx on the left side of the equation. To do this, we move the constant term, which is -21, to the right side of the equation. We accomplish this by adding 21 to both sides:

x2+4xβˆ’21+21=0+21x^2 + 4x - 21 + 21 = 0 + 21

This gives us:

x2+4x=21x^2 + 4x = 21

This step is essential because it prepares the equation for the creation of a perfect square trinomial on the left side. By moving the constant term to the right side, we create space to add a constant that will complete the square. This rearrangement allows us to focus on the x2x^2 and xx terms and manipulate them to form a perfect square. It is a necessary step in the process of completing the square and ensures that the equation is in the correct format for the subsequent steps.

Step 3: Complete the Square

Now, we complete the square. To do this, we take half of the coefficient of the xx term, square it, and add it to both sides of the equation. The coefficient of the xx term is 4. Half of 4 is 2, and 2 squared is 22=42^2 = 4. So, we add 4 to both sides:

x2+4x+4=21+4x^2 + 4x + 4 = 21 + 4

This simplifies to:

x2+4x+4=25x^2 + 4x + 4 = 25

The left side of the equation is now a perfect square trinomial. This is the core of the completing the square method. By adding the appropriate constant to both sides, we have transformed the left side into an expression that can be factored into the square of a binomial. This step is crucial because it allows us to rewrite the equation in a form that is easily solvable. The perfect square trinomial is the key to isolating xx and finding the solutions to the quadratic equation. The addition of the constant ensures that the equation remains balanced while creating the desired perfect square.

Step 4: Factor the Left Side

The left side of the equation, x2+4x+4x^2 + 4x + 4, is a perfect square trinomial and can be factored into (x+2)2(x + 2)^2. So, we rewrite the equation as:

(x+2)2=25(x + 2)^2 = 25

This factorization is a direct result of the previous step, where we completed the square. The perfect square trinomial we created is now expressed as the square of a binomial, which simplifies the equation significantly. This step is crucial because it sets up the equation for the final steps of solving for xx. By factoring the left side, we transform the equation into a form where the squared term is isolated, making it possible to apply the square root property and find the solutions.

Step 5: Take the Square Root of Both Sides

To solve for xx, we take the square root of both sides of the equation:

{sqrt{(x + 2)^2} = \pm\sqrt{25}}$ This gives us: $x + 2 = \pm 5$ Taking the square root of both sides introduces the $\pm$ sign, which is essential for capturing both possible solutions of the quadratic equation. The square root property is a fundamental tool in solving equations where a variable is squared. By applying this property, we eliminate the square and isolate the binomial containing $x$. This step is a critical bridge between the factored form of the equation and the individual solutions for $x$. It highlights the importance of considering both positive and negative roots when solving quadratic equations. ### Step 6: Solve for x Finally, we solve for $x$ by subtracting 2 from both sides of the equation: $x = -2 \pm 5$ This gives us two possible solutions for $x$: $x = -2 + 5 = 3$ $x = -2 - 5 = -7$ Thus, the solutions are $x = 3$ and $x = -7$. This final step involves isolating $x$ to obtain the solutions to the quadratic equation. By subtracting 2 from both sides, we separate $x$ and reveal the two values that satisfy the original equation. These solutions represent the points where the parabola intersects the x-axis. Finding these solutions is the ultimate goal of solving the quadratic equation, and this step completes the process of completing the square. The two distinct solutions highlight the nature of quadratic equations, which often have two roots due to the parabolic shape of their graphs. ## Conclusion By **completing the square**, we have successfully solved the quadratic equation $4x^2 + 16x - 84 = 0$. The solutions are $x = 3$ and $x = -7$. Therefore, the correct answer is **C) $x = -7, x = 3$**. This method not only provides the solutions but also demonstrates a powerful technique for manipulating and understanding quadratic equations. Mastering completing the square is an invaluable skill for anyone studying algebra and beyond. It provides a deep understanding of quadratic functions and their properties, making it an essential tool in mathematics. ## Final Answer: The final answer is $\boxed{x=-7, x=3}$ ##