Solving Quadratic Equations By Completing The Square A Step-by-Step Guide

Solving quadratic equations can sometimes feel like navigating a complex maze, but with the right strategies, it can become a rewarding journey. In this comprehensive guide, we will delve into the method of completing the square, a powerful technique for finding the solutions (also known as roots) of quadratic equations. We'll use a specific example to illustrate each step, ensuring you grasp the process thoroughly. Our focus will be on understanding the initial steps Gio should take to solve the equation $5x^2 + 15x - 4 = 0$ by completing the square.

Understanding Quadratic Equations

Before we dive into the specifics, let's establish a clear understanding of what quadratic equations are. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations arise in various fields, including physics, engineering, and economics, making their solutions crucial in many real-world applications. In our case, the equation $5x^2 + 15x - 4 = 0$ perfectly fits this definition, with a = 5, b = 15, and c = -4.

Why Completing the Square?

You might wonder, with other methods like factoring and the quadratic formula available, why should we focus on completing the square? While factoring is efficient for certain quadratic equations, it isn't always straightforward, especially when the roots are not integers or simple fractions. The quadratic formula, a universal solution, can sometimes feel like a black box. Completing the square, on the other hand, provides a step-by-step method that not only leads to the solution but also deepens your understanding of the structure of quadratic equations. It transforms the equation into a form where the variable appears inside a perfect square, making it easier to isolate and solve for 'x'. Furthermore, completing the square is the foundation upon which the quadratic formula itself is derived, highlighting its fundamental importance. This method is particularly useful when the quadratic equation cannot be easily factored, and it provides a clear, logical pathway to the solution. Understanding this method enhances problem-solving skills and provides a solid foundation for more advanced mathematical concepts.

Gio's First Step Isolate the Constant

When Gio is faced with the quadratic equation $5x^2 + 15x - 4 = 0$ and aims to solve it by completing the square, the very first strategic move is to isolate the constant term. This involves moving the constant term to the right side of the equation, setting the stage for the subsequent steps in the completing the square process. In our specific equation, the constant term is -4. To isolate it, Gio should add 4 to both sides of the equation. This action maintains the balance of the equation while segregating the variable terms on one side and the constant on the other. Mathematically, this step transforms the equation from $5x^2 + 15x - 4 = 0$ to $5x^2 + 15x = 4$. This seemingly simple step is crucial because it prepares the equation for the creation of a perfect square trinomial on the left-hand side, which is the core idea behind completing the square. By isolating the constant, we create a clearer pathway for manipulating the variable terms and transforming the quadratic expression into a more manageable form. This initial step is not just about moving a number; it's about strategically positioning the equation for success in the subsequent steps. Understanding the rationale behind this step is crucial for mastering the method of completing the square. Without isolating the constant, the subsequent steps become significantly more complicated, and the process of completing the square becomes much less intuitive. It's the foundation upon which the rest of the solution is built.

Why Not the Other Options?

Let's briefly discuss why the other options presented are not the correct first step. Option A suggests adding 9 to both sides, but this action lacks a clear purpose at this stage. Adding an arbitrary number doesn't directly contribute to completing the square. Option B, factoring 5 out of the variable terms, is a valid step later in the process, but it's premature as the initial action. Factoring out the leading coefficient is essential for creating a perfect square trinomial, but it should be done after isolating the constant. Option D, isolating the variable, is a vague instruction. We need to be more specific about which terms to isolate and how. Isolating the constant provides a clear direction and sets up the next steps logically. Therefore, isolating the constant is the most strategic and logical first step in solving the quadratic equation by completing the square.

Second Step Factor out the Leading Coefficient

After isolating the constant term, the subsequent crucial step in solving the quadratic equation $5x^2 + 15x = 4$ by completing the square is to factor out the leading coefficient from the variable terms. In this equation, the leading coefficient is 5, which is the coefficient of the $x^2$ term. Factoring out this coefficient is essential because the method of completing the square relies on having a leading coefficient of 1 for the $x^2$ term. By factoring out 5 from the terms on the left side of the equation, we transform the expression into a form that is amenable to completing the square. This involves dividing both the $5x^2$ and $15x$ terms by 5, resulting in the equation $5(x^2 + 3x) = 4$. This step is more than just algebraic manipulation; it's a strategic move that simplifies the process of creating a perfect square trinomial. When the leading coefficient is 1, we can directly apply the technique of adding and subtracting a specific value to complete the square. Factoring out the leading coefficient ensures that the coefficient of $x^2$ is 1, which is a prerequisite for the standard method of completing the square. This step clarifies the structure of the quadratic expression and prepares it for the next phase of the solution. Without this step, the process of completing the square becomes significantly more challenging and prone to errors. Understanding the rationale behind this step is key to mastering the method of completing the square. It's about setting up the equation in a way that allows for the easy application of the core principle of completing the square. It's a necessary bridge between the initial form of the equation and the creation of a perfect square trinomial.

Third Step Completing the Square

With the equation now in the form $5(x^2 + 3x) = 4$, the heart of the completing the square method comes into play. This involves transforming the expression inside the parentheses, $x^2 + 3x$, into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as $(x + a)^2$ or $(x - a)^2$. To achieve this transformation, we need to add a specific constant term to the expression $x^2 + 3x$. This constant is determined by taking half of the coefficient of the 'x' term (which is 3 in this case), squaring it, and adding the result inside the parentheses. Half of 3 is 3/2, and squaring it gives us $(3/2)^2 = 9/4$. This is the value we need to add to complete the square. However, we must remember that we're adding this value inside the parentheses, which is being multiplied by 5 on the left side of the equation. To maintain the balance of the equation, we must also add 5 times 9/4 to the right side. Therefore, we add 5 * (9/4) = 45/4 to the right side. The equation now becomes $5(x^2 + 3x + 9/4) = 4 + 45/4$. The expression inside the parentheses, $x^2 + 3x + 9/4$, is now a perfect square trinomial and can be factored as $(x + 3/2)^2$. This transformation is the essence of completing the square. It allows us to rewrite the quadratic expression in a form where the variable appears inside a squared term, making it easier to isolate and solve for 'x'. The act of completing the square is a strategic maneuver that simplifies the equation and brings us closer to the solution. It's a powerful technique that reveals the underlying structure of the quadratic equation and facilitates its resolution. Understanding this step thoroughly is crucial for mastering the method of completing the square.

Fourth Step Solve for x

Having successfully completed the square, the equation now stands as $5(x + 3/2)^2 = 4 + 45/4$. The next phase involves solving for 'x'. First, simplify the right side of the equation: 4 can be rewritten as 16/4, so $4 + 45/4 = 16/4 + 45/4 = 61/4$. The equation now reads $5(x + 3/2)^2 = 61/4$. To isolate the squared term, divide both sides of the equation by 5, yielding $(x + 3/2)^2 = 61/20$. The next step is to take the square root of both sides of the equation. Remember that when taking the square root, we must consider both positive and negative roots. This gives us $x + 3/2 = ±√(61/20)$. To isolate 'x', subtract 3/2 from both sides: $x = -3/2 ± √(61/20)$. This is the general solution for 'x'. To obtain numerical approximations, we can evaluate the square root term. The square root of 61/20 can be simplified as $√(61) / √(20)$, which is approximately $√(61) / (2√5)$. Further simplification and approximation can be done using a calculator. The final solutions for 'x' will be two distinct values, one obtained by adding the square root term and the other by subtracting it. These solutions represent the roots of the original quadratic equation. Solving for 'x' is the culmination of the completing the square process. It's the final step that reveals the values of 'x' that satisfy the equation. This process demonstrates the power and versatility of completing the square as a method for solving quadratic equations. It's a step-by-step approach that leads to a clear and accurate solution.

Conclusion

In summary, when Gio embarks on solving the quadratic equation $5x^2 + 15x - 4 = 0$ by completing the square, the crucial first step is to isolate the constant term. This involves adding 4 to both sides of the equation, resulting in $5x^2 + 15x = 4$. This foundational step sets the stage for the subsequent steps in the completing the square process. By understanding the rationale behind this step and mastering the subsequent techniques, Gio and anyone tackling quadratic equations can confidently navigate the solution and arrive at the correct answers. Completing the square is not just a method; it's a journey into the heart of quadratic equations, revealing their structure and solutions in a clear and logical way.