Completing The Square First Step Solving X^2-x-3=0

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 this article, we will guide Bao Yu (and anyone else struggling with this concept) through the process of completing the square, using the example equation $x^2 - x - 3 = 0$. We'll break down the initial steps and explain the reasoning behind them, ensuring a clear understanding of this valuable algebraic method.

Understanding the Problem

Bao Yu is faced with the quadratic equation $x^2 - x - 3 = 0$ and needs to solve it by completing the square. The question asks for the first step in this process. Let's analyze the options provided:

• A. Adding $-x$ to both sides of the equation
• B. Adding $-3$ to both sides of the equation
• C. Isolating the first term

To correctly identify the first step, we need to understand the goal of completing the square: to rewrite the quadratic equation in the form $(x + a)^2 = b$, where $a$ and $b$ are constants. This form allows us to easily solve for $x$ by taking the square root of both sides.

The First Step: Isolating the Constant Term

The initial step in completing the square involves isolating the constant term on the right side of the equation. This means moving the constant term (in this case, -3) to the other side of the equation. The primary reason for isolating the constant term in completing the square lies in setting up the equation for the subsequent steps, where we will manipulate the left side to form a perfect square trinomial. By moving the constant to the right side, we create space on the left side to add a specific value that will complete the square. This value is determined by taking half of the coefficient of the x-term, squaring it, and adding it to both sides of the equation, thus maintaining the balance. Isolating the constant term allows us to focus on transforming the quadratic expression into a perfect square, which is the core of the completing the square method. This process simplifies the equation into a form that can be easily solved by taking the square root of both sides, leading to the solutions for x. Think of it as preparing the canvas before you start painting; you need a clear space to work with.

To do this, we perform the opposite operation of subtraction, which is addition. Therefore, we add 3 to both sides of the equation:

$x^2 - x - 3 + 3 = 0 + 3$

This simplifies to:

$x^2 - x = 3$

This manipulation sets the stage for the next crucial step: completing the square on the left-hand side.

Why This Step is Crucial

Isolating the constant term is not just an arbitrary step; it's a fundamental preparation for the core of the completing the square method. By moving the constant to the right side, we create the necessary space on the left side to manipulate the quadratic expression 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 $(x - a)^2$. When we isolate the constant term, we can strategically determine a value to add to both sides of the equation. This value, derived from halving the coefficient of the x-term and squaring it, completes the square on the left side, allowing us to rewrite the quadratic equation in a solvable form. Without isolating the constant term, this critical manipulation becomes significantly more complex, hindering the completion of the square and making it difficult to find the solutions for x. Therefore, this initial step is not just about rearranging terms; it's about setting up the entire solving process for success.

The Next Steps: Completing the Square

With the constant term isolated, we can now focus on completing the square. This involves determining what constant to add to both sides of the equation to create a perfect square trinomial on the left-hand side. A perfect square trinomial can be factored into the form $(x + a)^2$ or $(x - a)^2$.

To find this constant, we take half of the coefficient of the $x$ term (which is -1 in this case), square it, and add it to both sides. Half of -1 is -1/2, and squaring it gives us $(-1/2)^2 = 1/4$.

So, we add 1/4 to both sides of the equation:

$x^2 - x + 1/4 = 3 + 1/4$

The left side can now be factored as a perfect square:

$(x - 1/2)^2 = 13/4$

Now, we can solve for $x$ by taking the square root of both sides:

$x - 1/2 = ±√(13/4)$

$x - 1/2 = ±√13 / 2$

Finally, we isolate $x$ by adding 1/2 to both sides:

$x = 1/2 ± √13 / 2$

Therefore, the solutions to the quadratic equation are:

$x = (1 + √13) / 2$ and $x = (1 - √13) / 2$

Why Option A and C are Incorrect

Let's briefly discuss why options A and C are not the correct first step:

• A. Adding $-x$ to both sides of the equation: Adding $-x$ to both sides would result in $x^2 - 3 = x$. While this is a valid algebraic manipulation, it doesn't move us closer to completing the square. It doesn't help in creating a perfect square trinomial on the left-hand side.
• C. Isolating the first term: Isolating the first term, $x^2$, would involve adding $x + 3$ to both sides, resulting in $x^2 = x + 3$. This, again, doesn't align with the goal of completing the square, which requires isolating the constant term first.

Both of these options complicate the process rather than simplifying it for completing the square.

Conclusion

Therefore, the correct first step for Bao Yu in completing the square to solve the equation $x^2 - x - 3 = 0$ is to isolate the constant term by adding 3 to both sides of the equation. This sets the stage for transforming the quadratic expression into a perfect square trinomial and ultimately finding the solutions for $x$. Mastering this initial step is crucial for successfully applying the completing the square method.

By understanding the reasoning behind each step, students like Bao Yu can confidently tackle quadratic equations and build a stronger foundation in algebra. Completing the square is not just a technique; it's a pathway to deeper algebraic understanding and problem-solving skills.

This method provides a structured approach to solving quadratic equations, offering a powerful tool in algebra. By focusing on isolating the constant term first, you set the stage for creating a perfect square trinomial, which simplifies the equation and leads to the solutions. Remember, practice is key to mastering this technique and building confidence in your mathematical abilities.