Solving $4z^2 + 11z - 3 = 0$ By Completing The Square: A Step-by-Step Guide

The quadratic equation $4z^2 + 11z - 3 = 0$ is a classic example that can be solved using a method known as completing the square. This method is not only a powerful tool for solving quadratic equations but also provides a deeper understanding of the structure and properties of these equations. In this comprehensive guide, we will walk through the step-by-step process of solving the given quadratic equation by completing the square, discuss the underlying principles, and highlight the advantages of this method. The ability to solve quadratic equations is a fundamental skill in algebra, with applications spanning various fields, including physics, engineering, and economics. Mastering the technique of completing the square provides a solid foundation for tackling more advanced mathematical concepts and problem-solving scenarios. By understanding this method, you gain not just a solution to a specific equation but also a versatile tool applicable to a wide range of mathematical problems. The process involves transforming the quadratic equation into a perfect square trinomial, which allows us to easily isolate the variable and find the solutions. This method is particularly useful when the quadratic equation cannot be easily factored or when we need to find the exact solutions, which may involve radicals. Moreover, completing the square is the basis for deriving the quadratic formula, a universal formula for solving any quadratic equation. This connection underscores the importance of mastering completing the square as it provides a foundational understanding of quadratic equations and their solutions. The technique involves algebraic manipulation to rewrite the quadratic expression in a form that includes a squared term, making it simpler to solve for the variable. This approach not only yields the solutions but also enhances your algebraic skills and problem-solving abilities, making it a valuable asset in your mathematical toolkit. Let's delve into the specifics of solving the equation $4z^2 + 11z - 3 = 0$ by completing the square, step by step, to ensure a clear and thorough understanding.

Step 1: Divide by the Leading Coefficient

Our initial equation is $4z^2 + 11z - 3 = 0$. To begin the process of completing the square, the first crucial step involves dividing the entire equation by the leading coefficient, which in this case is 4. This division ensures that the coefficient of the $z^2$ term becomes 1, simplifying the subsequent steps in the process. By dividing each term of the equation by 4, we transform the equation into a more manageable form. This step is essential because completing the square requires the quadratic expression to have a leading coefficient of 1. When the leading coefficient is not 1, the process of completing the square becomes more complex. Dividing by the leading coefficient allows us to apply the standard procedure for completing the square, making the process more straightforward and less prone to errors. After dividing by the leading coefficient, the equation takes on a form that is easier to manipulate and work with. This simplifies the identification of the terms needed to complete the square and sets the stage for the subsequent algebraic manipulations. It also helps in visualizing the structure of the quadratic expression and the relationships between its terms. The result of this division is a new equation where the coefficient of the squared term is unity, paving the way for the next steps in the process. This initial step is fundamental to the method of completing the square and is a critical prerequisite for the following operations. The transformed equation is now in a format that allows us to focus on completing the square by adding and subtracting the appropriate constant term. This step ensures that we can create a perfect square trinomial, which is a key component of the method. Dividing by the leading coefficient not only simplifies the equation but also sets the foundation for a systematic approach to finding the solutions. It is a critical step in ensuring the accuracy and efficiency of the completing the square method. The new equation is:

$z^2 + \frac{11}{4}z - \frac{3}{4} = 0$

Step 2: Move the Constant Term to the Right Side

Isolating the variable terms is the next key step in completing the square involves moving the constant term to the right side of the equation. In our transformed equation, $z^2 + \frac{11}{4}z - \frac{3}{4} = 0$, the constant term is $-\frac{3}{4}$. To move this term to the right side, we add $\frac{3}{4}$ to both sides of the equation. This operation isolates the terms containing the variable $z$ on the left side, preparing the equation for the completion of the square. Moving the constant term to the right side is a crucial step because it allows us to focus on manipulating the terms involving the variable to form a perfect square trinomial. This rearrangement makes it easier to identify the value needed to complete the square, which is the next step in the process. By isolating the variable terms, we create a clearer picture of what needs to be added to both sides of the equation to achieve a perfect square on the left. This step is essential for the algebraic manipulation required to complete the square. The process of moving the constant term involves a simple addition operation, but it is a significant step in transforming the equation into a more manageable form. It sets the stage for adding a specific value that will allow us to rewrite the left side as a squared binomial. Isolating the variable terms ensures that the next steps can be performed with greater clarity and accuracy. This step is not just a rearrangement; it is a strategic move that brings us closer to the solution by facilitating the creation of a perfect square trinomial. The equation now becomes:

$z^2 + \frac{11}{4}z = \frac{3}{4}$

Step 3: Complete the Square

To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(az + b)^2$ or $(az - b)^2$. The value we need to add is determined by taking half of the coefficient of the $z$ term, squaring it, and adding it to both sides of the equation. In our equation, $z^2 + \frac{11}{4}z = \frac{3}{4}$, the coefficient of the $z$ term is $\frac{11}{4}$. Half of this coefficient is $\frac{11}{8}$, and squaring it gives us $(\frac{11}{8})^2 = \frac{121}{64}$. Therefore, we add $\frac{121}{64}$ to both sides of the equation. This step is the heart of the completing the square method, as it transforms the left side of the equation into a perfect square trinomial. Adding the correct value is essential for the subsequent steps, as it allows us to rewrite the left side as a squared binomial, which simplifies the process of solving for the variable. The addition of this value maintains the balance of the equation while simultaneously creating a perfect square on the left side. This step is not just a mathematical manipulation; it is a strategic transformation that brings us closer to isolating the variable and finding the solutions. Completing the square is a powerful technique that enables us to solve quadratic equations by rewriting them in a more manageable form. This method is particularly useful when the quadratic equation cannot be easily factored or when we need to find exact solutions. The process involves algebraic manipulation to rewrite the quadratic expression in a form that includes a squared term, making it simpler to solve for the variable. This step ensures that we can apply the square root property to solve for $z$ in the next step. The new equation is:

$z^2 + \frac{11}{4}z + \frac{121}{64} = \frac{3}{4} + \frac{121}{64}$

Step 4: Factor the Left Side and Simplify the Right Side

Factoring and simplifying is the step in the process involves factoring the left side of the equation, which is now a perfect square trinomial, and simplifying the right side. The left side, $z^2 + \frac{11}{4}z + \frac{121}{64}$, can be factored into $(z + \frac{11}{8})^2$. This factorization is a direct result of completing the square in the previous step. The right side of the equation, $\frac{3}{4} + \frac{121}{64}$, needs to be simplified by finding a common denominator and adding the fractions. The common denominator for 4 and 64 is 64. Converting $\frac{3}{4}$ to a fraction with a denominator of 64, we get $\frac{48}{64}$. Adding this to $\frac{121}{64}$ gives us $\frac{169}{64}$. Factoring the left side and simplifying the right side are essential steps in solving the quadratic equation. Factoring the left side allows us to rewrite the equation in a form where we can easily isolate the variable. Simplifying the right side ensures that we have a single numerical value, which is necessary for the subsequent steps. This step is a crucial bridge between completing the square and finding the solutions for $z$. The ability to factor the perfect square trinomial is a direct consequence of the careful steps taken in completing the square. Simplifying the right side ensures that the equation is in its most manageable form before we proceed to isolate the variable. These operations are fundamental algebraic manipulations that are key to solving quadratic equations using the completing the square method. The process of factoring and simplifying is a testament to the power of algebraic techniques in transforming complex equations into simpler forms. This step sets the stage for the final steps in solving for the variable. The equation now becomes:

$(z + \frac{11}{8})^2 = \frac{169}{64}$

Step 5: Take the Square Root of Both Sides

Applying the square root property is a crucial step towards isolating the variable $z$. In the equation $(z + \frac{11}{8})^2 = \frac{169}{64}$, we take the square root of both sides. This operation is based on the principle that if $a^2 = b$, then $a = \pm\sqrt{b}$. Applying this to our equation, we get $z + \frac{11}{8} = \pm\sqrt{\frac{169}{64}}$. The square root of $\frac{169}{64}$ is $\frac{13}{8}$, so we have $z + \frac{11}{8} = \pm\frac{13}{8}$. Taking the square root of both sides is a fundamental step in solving equations involving squared terms. It allows us to remove the square and bring the variable closer to isolation. The introduction of the $\pm$ sign is critical because it acknowledges that there are two possible square roots for any positive number: a positive and a negative root. This step ensures that we capture both potential solutions for the variable. Applying the square root property is a direct consequence of completing the square and factoring the left side. It is a key step in unraveling the equation and isolating the variable. This step highlights the importance of understanding the properties of square roots and their role in solving algebraic equations. The process of taking the square root of both sides is a powerful technique that simplifies the equation and brings us closer to the final solutions. It is a critical step in the completing the square method and is essential for finding the values of $z$. The equation now splits into two separate equations:

$z + \frac{11}{8} = \frac{13}{8}$ and $z + \frac{11}{8} = -\frac{13}{8}$

Step 6: Solve for z

Isolating the variable z to find the solutions. We have two equations: $z + \frac{11}{8} = \frac{13}{8}$ and $z + \frac{11}{8} = -\frac{13}{8}$. To solve for $z$ in each equation, we subtract $\frac{11}{8}$ from both sides. For the first equation, $z + \frac{11}{8} = \frac{13}{8}$, subtracting $\frac{11}{8}$ from both sides gives us $z = \frac{13}{8} - \frac{11}{8} = \frac{2}{8} = \frac{1}{4}$. For the second equation, $z + \frac{11}{8} = -\frac{13}{8}$, subtracting $\frac{11}{8}$ from both sides gives us $z = -\frac{13}{8} - \frac{11}{8} = -\frac{24}{8} = -3$. Therefore, the solutions for $z$ are $\frac{1}{4}$ and -3. Solving for $z$ is the final step in the process, where we isolate the variable to find its values. This step involves simple algebraic manipulations, such as adding or subtracting the same value from both sides of the equation. The goal is to get $z$ by itself on one side of the equation, which reveals its value. This step is a direct result of the previous steps, where we completed the square, factored the left side, and took the square root of both sides. Each of these steps prepared the equation for this final isolation of the variable. The process of solving for $z$ demonstrates the power of algebraic techniques in unraveling equations and finding solutions. It highlights the importance of understanding the basic principles of equation solving, such as maintaining balance by performing the same operation on both sides. Finding the solutions for $z$ completes the process of solving the quadratic equation. These solutions are the values of $z$ that make the original equation true. The ability to solve for $z$ is a fundamental skill in algebra and is essential for various applications in mathematics and other fields. The solutions we have found are the roots of the quadratic equation and represent the points where the parabola intersects the x-axis. Thus, the solutions for the quadratic equation $4z^2 + 11z - 3 = 0$ are:

$z = \frac{1}{4}$ and $z = -3$

Conclusion

In summary, we have successfully solved the quadratic equation $4z^2 + 11z - 3 = 0$ by completing the square. This method involves a series of algebraic manipulations, including dividing by the leading coefficient, moving the constant term, completing the square, factoring, taking the square root, and solving for the variable. The solutions we found are $z = \frac{1}{4}$ and $z = -3$. Completing the square is a powerful technique for solving quadratic equations, especially when they cannot be easily factored. It provides a systematic approach to finding the solutions and offers a deeper understanding of the structure of quadratic equations. This method is not only a valuable tool for solving equations but also serves as a foundation for more advanced mathematical concepts. The process of completing the square demonstrates the elegance and power of algebraic techniques in transforming and solving equations. Each step in the process is carefully designed to bring us closer to the solutions, and the final result is a testament to the effectiveness of the method. Understanding completing the square is essential for anyone studying algebra, as it provides a fundamental skill that is applicable in various mathematical contexts. This method is particularly useful when dealing with equations that do not factor easily or when we need to find exact solutions, which may involve radicals. Completing the square is also the basis for deriving the quadratic formula, a universal formula for solving any quadratic equation. This connection underscores the importance of mastering completing the square as it provides a foundational understanding of quadratic equations and their solutions. The ability to solve quadratic equations is a critical skill in mathematics, with applications spanning various fields, including physics, engineering, and economics. By mastering the technique of completing the square, you gain not just a solution to a specific equation but also a versatile tool applicable to a wide range of mathematical problems. This comprehensive guide has walked you through each step of the process, providing a clear and thorough understanding of how to solve quadratic equations by completing the square. With practice, this method will become a valuable asset in your mathematical toolkit. The solutions we have found represent the points where the parabola defined by the quadratic equation intersects the x-axis. This geometric interpretation adds another layer of understanding to the solutions and their significance.