Next Step For Solving Quadratic Equation By Completing The Square
In this comprehensive guide, we will delve into the process of solving a quadratic equation by completing the square. We will meticulously analyze the steps taken so far and determine the next crucial step in arriving at the solution. Our focus will be on the given equation: $-3=-2 x^2+2 x$. We'll break down each stage, ensuring a clear understanding of the underlying principles and techniques involved. This detailed explanation will not only help you solve this particular equation but also equip you with the knowledge to tackle similar problems with confidence. We will explore the logic behind each manipulation and highlight common pitfalls to avoid, making this a valuable resource for anyone seeking to master the method of completing the square.
1. Transforming the Equation: $-3=-2 x^2+2 x$ to $0=-2 x^2+2 x+3$
The initial step in solving any quadratic equation is to bring all terms to one side, setting the equation equal to zero. This standard form, $ax^2 + bx + c = 0$, is essential for applying various solution methods, including completing the square. In our case, we start with $-3=-2 x^2+2 x$. To achieve the standard form, we add 3 to both sides of the equation. This operation maintains the equality while rearranging the terms. By adding 3 to both sides, the -3 on the left side cancels out, leaving us with zero. On the right side, we add 3 to the expression $-2 x^2+2 x$, resulting in $-2 x^2+2 x + 3$. This transformation is a fundamental algebraic manipulation that preserves the equation's integrity. Understanding this step is crucial because it sets the stage for subsequent operations. The goal is to have the quadratic expression on one side and zero on the other, preparing the equation for further analysis and solution. This rearrangement is not just a cosmetic change; it's a necessary prerequisite for applying the completing the square technique effectively. The equation $0=-2 x^2+2 x+3$ now represents the same relationship as the original equation but in a format that is more conducive to solving. This step highlights the importance of algebraic manipulation in simplifying equations and making them amenable to specific solution methods. By recognizing the need for the standard form, we lay the groundwork for a systematic approach to solving quadratic equations.
2. Isolating the Quadratic and Linear Terms: $-3=-2(x^2-x)$
After setting the equation to the standard form, the next step in completing the square involves isolating the quadratic and linear terms. This means focusing on the terms containing $x^2$ and $x$, while moving the constant term to the other side of the equation. However, in the given sequence, there's a slight deviation from the typical process. Instead of directly isolating the constant term, the equation proceeds by factoring out the coefficient of the $x^2$ term from the right side of the original equation, $-3=-2 x^2+2 x$. This step is crucial because completing the square works most effectively when the coefficient of $x^2$ is 1. By factoring out -2 from the terms $-2x^2$ and $2x$, we obtain $-2(x^2-x)$. This manipulation is achieved by dividing each term by -2. Dividing $-2x^2$ by -2 yields $x^2$, and dividing $2x$ by -2 results in $-x$. The equation now transforms to $-3=-2(x^2-x)$. This step is essential because it simplifies the expression inside the parentheses, making it easier to complete the square. Factoring out the leading coefficient prepares the quadratic expression for the next stage, where we will add a constant term to create a perfect square trinomial. Understanding this step is vital as it directly influences the success of the completing the square method. It showcases the importance of strategic algebraic manipulation in simplifying complex expressions and paving the way for efficient problem-solving. By isolating the quadratic and linear terms in this manner, we are setting the stage for the core process of completing the square.
3. Preparing to Complete the Square: $-3+\square=-2(x^2-x+\square)$
Now that we have the equation in the form $-3=-2(x^2-x)$, the next step is to prepare for completing the square. This involves adding a specific value inside the parentheses to create a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored into the form $(x + a)^2$ or $(x - a)^2$. To achieve this, we need to add a constant term that makes the expression inside the parentheses a perfect square. However, it's crucial to remember that whatever we add inside the parentheses on one side of the equation, we must also account for on the other side to maintain the equality. The equation $-3+\square=-2(x^2-x+\square)$ explicitly shows that we are adding a value to both sides. The squares () represent the values that need to be determined and added. The key to finding this value lies in the coefficient of the x term inside the parentheses. In our case, the coefficient of x is -1. To complete the square, we take half of this coefficient, square it, and add the result inside the parentheses. Half of -1 is -1/2, and squaring -1/2 gives us 1/4. So, we need to add 1/4 inside the parentheses. However, because the parentheses are multiplied by -2 on the right side of the equation, adding 1/4 inside the parentheses actually means we are adding -2 * (1/4) = -1/2 to the right side of the equation. Therefore, we must add -1/2 to the left side as well to maintain the balance. This step underscores the importance of careful consideration of the equation's structure when manipulating terms. It's not just about adding the same number to both sides, but also accounting for any coefficients that may affect the value being added or subtracted. By preparing the equation in this way, we are setting the stage for the final steps of completing the square, where we will factor the perfect square trinomial and solve for x.
4. Determining the Next Step: Adding the Correct Value to Complete the Square
Having prepared the equation as $-3+\square=-2(x^2-x+\square)$, the next critical step is to determine the value that completes the square. As discussed in the previous section, this involves calculating the constant term that, when added to the expression inside the parentheses, will transform it into a perfect square trinomial. This constant is found by taking half of the coefficient of the x term, squaring it, and then adding the result. In our equation, the expression inside the parentheses is $x^2 - x$. The coefficient of the x term is -1. Half of -1 is -1/2, and squaring -1/2 gives us 1/4. Therefore, the value we need to add inside the parentheses is 1/4. However, we must also account for the -2 outside the parentheses. Adding 1/4 inside the parentheses is equivalent to adding -2 * (1/4) = -1/2 to the right side of the equation. To maintain the balance, we must also add -1/2 to the left side of the equation. So, the next step is to add -1/2 to both sides of the equation, specifically: $-3 + (-1/2) = -2(x^2 - x + 1/4)$. This step is paramount because it directly leads to the formation of a perfect square trinomial. By adding the correct value, we transform the expression inside the parentheses into a form that can be easily factored as a squared binomial. This is the heart of the completing the square method, as it allows us to rewrite the quadratic equation in a more manageable form. The equation now becomes $-7/2 = -2(x - 1/2)^2$. From here, we can proceed to solve for x by isolating the squared term, taking the square root, and then solving the resulting linear equations. This step highlights the precision required in algebraic manipulations and the importance of understanding the underlying principles of completing the square. By correctly determining and adding the constant term, we pave the way for a straightforward solution to the quadratic equation.
In summary, the next step for solving the quadratic equation by completing the square is to add -1/2 to both sides of the equation, resulting in $-3 + (-1/2) = -2(x^2 - x + 1/4)$. This crucial step completes the square, transforming the quadratic expression into a perfect square trinomial, which can then be factored and solved for x.
