Steps In Completing The Square To Solve 3 - 4x = 5x² - 14x
Completing the square is a powerful technique for solving quadratic equations. It transforms a quadratic equation into a perfect square trinomial, making it easier to isolate the variable and find the solutions. In the equation $3 - 4x = 5x^2 - 14x$, we need to follow specific steps to successfully apply this method. This article aims to detail the steps in completing the square and provide a comprehensive understanding of how to solve the given quadratic equation. We will explore each step in detail, ensuring clarity and accuracy in the solution process. Mastering this technique is essential for anyone studying algebra, as it not only helps in solving quadratic equations but also lays the groundwork for understanding more advanced mathematical concepts.
Understanding the Importance of Completing the Square
Before diving into the specific steps, let's understand why completing the square is so important. This method is not just a way to solve equations; it provides insight into the structure of quadratic functions. It helps us rewrite the quadratic equation in vertex form, which reveals the vertex of the parabola represented by the equation. The vertex is a critical point as it represents the maximum or minimum value of the quadratic function. Furthermore, completing the square is a foundational concept for deriving the quadratic formula, a universal tool for solving any quadratic equation. By understanding the process of completing the square, we gain a deeper appreciation for the underlying principles of algebra and enhance our problem-solving skills. This technique is also widely used in various fields, including physics, engineering, and economics, where quadratic equations often arise in modeling real-world phenomena.
Step-by-Step Guide to Completing the Square for 3 - 4x = 5x² - 14x
To solve the equation $3 - 4x = 5x^2 - 14x$ by completing the square, we will follow these steps:
1. Rearrange the Equation
Our first goal is to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. To achieve this, we need to move all terms to one side of the equation. Starting with $3 - 4x = 5x^2 - 14x$, we can subtract $3$ and add $4x$ to both sides. This gives us: $0 = 5x^2 - 14x + 4x - 3$. Simplifying further, we combine the $x$ terms: $0 = 5x^2 - 10x - 3$. This rearranged form sets the stage for the subsequent steps in completing the square. Ensuring the equation is in this standard form is crucial because it allows us to clearly identify the coefficients $a$, $b$, and $c$, which are essential for the next steps. A common mistake is to skip this step or to incorrectly rearrange the terms, which can lead to errors in the final solution. Therefore, careful attention to this initial step is vital for the successful application of the completing the square method.
2. Factor out the Leading Coefficient
Next, we need to ensure that the coefficient of the $x^2$ term is $1$. In our equation, $0 = 5x^2 - 10x - 3$, the leading coefficient is $5$. To make it $1$, we factor out $5$ from the terms containing $x$. This gives us: $0 = 5(x^2 - 2x) - 3$. Notice that we only factor out the $5$ from the terms with $x$, leaving the constant term $-3$ outside the parentheses. This step is crucial because completing the square works most effectively when the leading coefficient is $1$. Factoring out the leading coefficient allows us to focus on the quadratic expression inside the parentheses, making the subsequent steps more manageable. It's important to perform this step accurately, as any error here will propagate through the rest of the solution. Many students find this step challenging, but with practice, it becomes a straightforward part of the completing the square process. Remember, the goal is to simplify the quadratic expression inside the parentheses so that we can easily create a perfect square trinomial.
3. Complete the Square
This is the core of the method. Inside the parentheses, we have the expression $x^2 - 2x$. To complete the square, we need to add and subtract a value that will turn this expression into a perfect square trinomial. The value we need to add is determined by taking half of the coefficient of the $x$ term, squaring it. In this case, the coefficient of the $x$ term is $-2$. Half of $-2$ is $-1$, and squaring $-1$ gives us $1$. So, we add and subtract $1$ inside the parentheses: $0 = 5(x^2 - 2x + 1 - 1) - 3$. Notice that we are adding and subtracting the same value, which doesn't change the overall value of the expression. Now, we can rewrite the first three terms inside the parentheses as a perfect square: $0 = 5((x - 1)^2 - 1) - 3$. The expression $x^2 - 2x + 1$ is equivalent to $(x - 1)^2$. This is the essence of completing the square – transforming a quadratic expression into a perfect square trinomial, which can be factored into a binomial squared. This step requires a solid understanding of algebraic manipulation and the properties of perfect square trinomials. It's a crucial step in the process, as it sets up the equation for solving by isolating the variable.
4. Distribute and Simplify
Now, we distribute the $5$ back into the parentheses: $0 = 5(x - 1)^2 - 5 - 3$. This gives us: $0 = 5(x - 1)^2 - 8$. This step is necessary to combine the constant terms and isolate the squared term. Distributing the leading coefficient ensures that all terms are properly accounted for in the equation. It's a straightforward algebraic step, but it's essential to perform it accurately to avoid errors. Many students make mistakes by forgetting to distribute the coefficient to all terms inside the parentheses. After distributing, we simplify the equation by combining the constant terms. This simplification makes the equation easier to solve in the subsequent steps. This step is a crucial bridge between completing the square and isolating the variable to find the solutions.
5. Isolate the Squared Term
Our next step is to isolate the squared term. To do this, we add $8$ to both sides of the equation: $8 = 5(x - 1)^2$. Then, we divide both sides by $5$: $\frac{8}{5} = (x - 1)^2$. This step brings us closer to solving for $x$ by getting the squared term alone on one side of the equation. Isolating the squared term is a key step in solving equations involving squares, as it allows us to take the square root of both sides in the next step. This process requires careful algebraic manipulation to ensure that the equation remains balanced. It's important to perform these operations correctly to avoid introducing errors. By isolating the squared term, we set the stage for the final steps in finding the values of $x$ that satisfy the equation.
6. Take the Square Root
To eliminate the square, we take the square root of both sides of the equation: $\pm \sqrt{\frac{8}{5}} = x - 1$. Remember to include both the positive and negative square roots, as both will yield valid solutions. Taking the square root is the inverse operation of squaring, and it allows us to remove the square from the binomial. The inclusion of both positive and negative roots is crucial because both values, when squared, will result in the same positive value. This step is a critical juncture in the solution process, as it leads us to the final isolation of the variable $x$. It's important to be meticulous in this step, ensuring that both possible roots are considered to obtain a complete set of solutions.
7. Solve for x
Finally, we solve for $x$ by adding $1$ to both sides: $x = 1 \pm \sqrt\frac{8}{5}}$. We can simplify the square root further by rationalizing the denominator{5}} = 1 \pm \sqrt{\frac{8 \cdot 5}{5 \cdot 5}} = 1 \pm \sqrt{\frac{40}{25}} = 1 \pm \frac{\sqrt{40}}{5} = 1 \pm \frac{2\sqrt{10}}{5}$. Thus, the solutions are $x = 1 + \frac{2\sqrt{10}}{5}$ and $x = 1 - \frac{2\sqrt{10}}{5}$. This final step involves isolating the variable $x$ to find the solutions to the original equation. It often involves additional algebraic manipulations, such as rationalizing the denominator, to express the solutions in simplest form. Solving for $x$ is the culmination of the completing the square process, and it provides the values that satisfy the original quadratic equation. These solutions represent the points where the parabola intersects the x-axis, and they are critical for understanding the behavior of the quadratic function.
Conclusion
Completing the square is a valuable technique for solving quadratic equations. By following these steps carefully, we can transform any quadratic equation into a solvable form. The key steps include rearranging the equation, factoring out the leading coefficient, completing the square, distributing and simplifying, isolating the squared term, taking the square root, and solving for $x$. Mastering this method provides a solid foundation for more advanced algebraic concepts and problem-solving skills. Understanding completing the square steps is not just about finding solutions; it's about understanding the structure and properties of quadratic equations. This knowledge empowers us to tackle a wide range of mathematical problems and applications in various fields. By practicing these steps and understanding the underlying principles, we can confidently solve quadratic equations and enhance our mathematical proficiency. The ability to complete the square is a testament to one's algebraic skills and provides a pathway to deeper mathematical understanding.
