Completing The Square: First Step For $x^2-x-3=0$

When you're faced with solving a polynomial equation, especially one like $x^2-x-3=0$, using techniques like completing the square can be incredibly powerful. But what's the very first step you should take to set yourself up for success with this method? Let's dive into the initial move that gets the ball rolling. The core idea behind completing the square is to manipulate the equation so that one side becomes a perfect square trinomial, which can then be easily factored. To achieve this, we need to isolate the terms involving $x^2$ and $x$ on one side of the equation, leaving the constant term on the other. Therefore, the first step to solve the polynomial equation $x^2-x-3=0$ by completing the square is to isolate the constant, -3. This means moving the constant term to the right side of the equation, transforming $x^2-x-3=0$ into $x^2-x = 3$. Why is this crucial? Because the subsequent steps involve adding a specific value to both sides of the equation to create that perfect square trinomial. If the constant is still on the same side as the $x^2$ and $x$ terms, it complicates the process and makes it harder to see how to proceed. Options A, B, and C are not the most direct or effective first steps. Adding $-x$ to both sides (A) doesn't help create a perfect square. Adding -3 to both sides (B) would result in $x^2-x-6=0$, which doesn't isolate the necessary components for completing the square. Isolating the first term, $x^2$ (C), is also not the primary goal at this stage; we want to group the variable terms together. So, always start by getting that constant term out of the way!

Understanding the 'Why' Behind Isolating the Constant

Let's elaborate on why isolating the constant term is the foundational step when employing the completing the square method. The objective of completing the square is to transform an expression of the form $ax^2 + bx + c = 0$ into a form where $x$ can be easily solved, typically $(x+h)^2 = k$. To achieve this structural change, we must first ensure that the terms containing the variable, $x^2$ and $x$, are on one side of the equation, and the standalone numerical value, the constant, is on the other. For our specific equation, $x^2-x-3=0$, the terms with variables are $x^2$ and $-x$, and the constant is $-3$. By moving $-3$ to the right side, we get $x^2-x = 3$. This arrangement is essential because the next critical step in completing the square involves adding a specific quantity to both sides of the equation. This quantity is calculated based on the coefficient of the $x$ term. Specifically, you take half of the coefficient of the $x$ term, square it, and add that result to both sides. In $x^2-x = 3$, the coefficient of the $x$ term is $-1$. Half of $-1$ is $-1/2$, and squaring that gives us $(-1/2)^2 = 1/4$. So, we would add $1/4$ to both sides: $x^2-x + 1/4 = 3 + 1/4$. The left side, $x^2-x+1/4$, is now a perfect square trinomial, specifically $(x-1/2)^2$. If we hadn't isolated the constant first, this addition step would look messy, potentially like $x^2-x-3 + 1/4 = 0 + 1/4$, which obscures the perfect square formation. Therefore, isolating the constant, -3, is not just a suggestion; it's a prerequisite for the standard and most effective execution of the completing the square method. It simplifies the algebra and makes the transformation into a solvable squared term straightforward.

Evaluating Other Potential First Steps

While we've established that isolating the constant is the key first step, it's worth examining why the other options presented (A, B, and C) are not the optimal starting points for solving $x^2-x-3=0$ by completing the square. Let's break them down:

• Option A: Adding $-x$ to both sides of the equation. If we were to add $-x$ to both sides of $x^2-x-3=0$, the equation would become $x^2-2x-3 = -x$. This operation moves a variable term and doesn't isolate the constant or group the variable terms effectively for completing the square. The goal is to get $x^2$ and $x$ terms together, and the constant separate. This step moves us further away from that objective.

• Option B: Adding -3 to both sides of the equation. This option might seem tempting because it involves the constant term. However, if we add $-3$ to both sides of $x^2-x-3=0$, we get $x^2-x-3-3 = 0-3$, which simplifies to $x^2-x-6=0$. This doesn't isolate the constant on one side; instead, it introduces another constant term to the left side and leaves 0 on the right. The completing the square method requires the constant to be moved to the opposite side, not simply added to both sides in a way that maintains it on the variable side. The target structure is $(x+h)^2 = k$, not $ax^2+bx+c=0$ with an adjusted constant.

• Option C: Isolating the first term, $x^2$. To isolate $x^2$ in $x^2-x-3=0$, we would need to move both $-x$ and $-3$ to the other side. This would result in $x^2 = x+3$. While this does place $x^2$ alone on one side, it doesn't set up the equation for completing the square as effectively as isolating the constant does. Completing the square focuses on creating a perfect trinomial from the $x^2$ and $x$ terms, and having the constant available on the right side is crucial for that process. $x^2 = x+3$ is a valid algebraic manipulation, but it's not the most direct pathway to completing the square.

In contrast, as we've seen, isolating the constant $-3$ (moving it to the right side to get $x^2-x=3$) prepares the equation perfectly for the next step: adding $(b/2)^2$ to both sides to create the perfect square trinomial. This strategic first move is what makes the rest of the completing the square process manageable and efficient.

The Completing the Square Process Step-by-Step

Now that we've established the crucial first step, let's walk through the entire process of solving $x^2-x-3=0$ by completing the square to solidify your understanding. This method is a fantastic way to solve quadratic equations, especially when factoring isn't obvious or possible.

Step 1: Isolate the Constant Term. As discussed extensively, our first move is to get the constant term away from the $x^2$ and $x$ terms. Original equation: $x^2-x-3=0$ Add 3 to both sides: $x^2-x = 3$ This sets the stage perfectly for the next crucial step.

Step 2: Prepare for the Perfect Square. Now, we need to make the left side of the equation a perfect square trinomial. We do this by adding a specific value to both sides. This value is found by taking the coefficient of the $x$ term, dividing it by 2, and then squaring the result. In our equation, $x^2-x = 3$, the coefficient of the $x$ term is $-1$. Half of $-1$ is $-1/2$. Squaring $-1/2$ gives us $(-1/2)^2 = 1/4$. So, we add $1/4$ to both sides of the equation: $x^2-x + 1/4 = 3 + 1/4$

Step 3: Factor the Perfect Square Trinomial. The left side of the equation, $x^2-x+1/4$, is now a perfect square trinomial. It can be factored into the form $(x+h)^2$ or $(x-h)^2$. In this case, it factors as $(x - 1/2)^2$. The value inside the parenthesis is always the coefficient of the $x$ term divided by 2 (which we calculated in Step 2). Our equation now looks like: $(x - 1/2)^2 = 3 + 1/4$

Step 4: Simplify the Right Side. Combine the terms on the right side of the equation. $3 + 1/4 = 12/4 + 1/4 = 13/4$ So, the equation becomes: $(x - 1/2)^2 = 13/4$

Step 5: Solve for x using the Square Root Property. Now that we have a squared term isolated, we can take the square root of both sides to solve for $x$. Remember that when you take the square root, you must consider both the positive and negative roots. $\sqrt{(x - 1/2)^2} = \pm\sqrt{13/4}$ $x - 1/2 = \pm\frac{\sqrt{13}}{2}$

Step 6: Isolate x. Finally, add $1/2$ to both sides of the equation to get $x$ by itself. $x = 1/2 \pm\frac{\sqrt{13}}{2}$ This gives us two distinct solutions: $x = \frac{1 + \sqrt{13}}{2}$ and $x = \frac{1 - \sqrt{13}}{2}$

By following these steps, with the crucial first step of isolating the constant, we've successfully solved the quadratic equation $x^2-x-3=0$ using the completing the square method. It's a robust technique that, once understood, becomes a go-to tool in your algebra arsenal.

In conclusion, the journey to solving $x^2-x-3=0$ via completing the square begins with a clear objective: to manipulate the equation into a form where a perfect square can be easily created. The first step to solve the polynomial equation $x^2-x-3=0$ by completing the square is to isolate the constant, -3, moving it to the right side to yield $x^2-x = 3$. This strategic move paves the way for adding the necessary term to create the perfect square trinomial, ultimately leading to the solutions. This method underscores the importance of understanding the underlying structure of quadratic equations and how algebraic manipulations can unlock their solutions. For more insights into quadratic equations and algebraic techniques, exploring resources like Khan Academy's Algebra Section can provide further depth and practice opportunities.