Solving Quadratic Equations By Completing The Square Maya's First Step

Introduction to Completing the Square

The method of completing the square is a powerful algebraic technique used to solve quadratic equations. This method transforms a quadratic equation into a perfect square trinomial, making it easier to find the solutions. Understanding each step is crucial for successfully applying this method. In this comprehensive guide, we will delve into the initial step Maya should take when solving the quadratic equation $4x^2 + 16x + 3 = 0$ by completing the square. We will break down the process and provide clear explanations to ensure you grasp the underlying concepts.

The Given Quadratic Equation

We are given the quadratic equation:

$4x^2 + 16x + 3 = 0$

Maya needs to solve this equation by completing the square. To begin, it's essential to understand the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. In our case, $a = 4$, $b = 16$, and $c = 3$. The goal of completing the square is to rewrite the equation in the form $(x + p)^2 = q$, where $p$ and $q$ are constants. This form allows us to easily solve for $x$ by taking the square root of both sides.

The First Crucial Step Isolate the Constant Term

The first step in completing the square is to isolate the constant term on one side of the equation. This means we need to move the constant term, which is 3 in our equation, to the right side. To do this, we subtract 3 from both sides of the equation. This ensures that the terms involving $x$ are on one side, and the constant term is on the other. Isolating the constant sets the stage for the subsequent steps in the process. Without this initial step, it would be significantly more challenging to proceed with completing the square effectively. Therefore, it is not just a procedural step, but a foundational element of the method.

Performing the Isolation

Subtracting 3 from both sides of the equation:

$4x^2 + 16x + 3 - 3 = 0 - 3$

This simplifies to:

$4x^2 + 16x = -3$

Now, we have the constant term isolated on the right side of the equation, which is a necessary condition for proceeding with completing the square.

Why Isolate the Constant First?

Isolating the constant term is crucial because it allows us to focus on the terms involving $x$ when completing the square. The process involves manipulating these terms to form a perfect square trinomial. By moving the constant term to the other side, we create space to add a specific value that completes the square. This value is determined by the coefficient of the $x$ term. If the constant term is not isolated, the process of completing the square becomes more complex and prone to errors. The clarity gained by isolating the constant makes the subsequent algebraic manipulations more straightforward and intuitive. This step is not arbitrary but rather a carefully designed part of the methodology to simplify the solution process.

Addressing Other Options

Let's briefly discuss why the other options provided are not the correct first step:

B. Subtracting 16x from Both Sides

Subtracting $16x$ from both sides would result in:

$4x^2 + 3 = -16x$

While this is a valid algebraic manipulation, it does not move us closer to completing the square. It actually complicates the process by mixing the $x^2$ and $x$ terms on different sides of the equation. Therefore, this is not the appropriate first step.

C. Isolating the Variable

Isolating the variable completely at this stage is not feasible. The equation has both $x^2$ and $x$ terms, so we cannot simply isolate $x$ as we would in a linear equation. Completing the square is specifically designed to handle quadratic equations where isolating the variable directly is not possible. Isolating the variable is the final goal, but it cannot be achieved in the first step.

Continuing the Process of Completing the Square

Now that Maya has correctly isolated the constant term, the next steps involve making the coefficient of $x^2$ equal to 1 and then completing the square. We will outline these steps briefly to provide a complete picture of the method.

Step 2: Divide by the Coefficient of x²

In our equation, the coefficient of $x^2$ is 4. To make it 1, we divide the entire equation by 4:

$(4x^2 + 16x) / 4 = -3 / 4$

This simplifies to:

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

Step 3: Complete the Square

To complete the square, we take half of the coefficient of the $x$ term, square it, and add it to both sides of the equation. The coefficient of the $x$ term is 4. Half of 4 is 2, and 2 squared is 4. So, we add 4 to both sides:

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

Step 4: Factor and Simplify

The left side of the equation is now a perfect square trinomial, which can be factored as:

$(x + 2)^2 = -3/4 + 16/4$

Simplifying the right side:

$(x + 2)^2 = 13/4$

Step 5: Solve for x

Now, we take the square root of both sides:

$x + 2 = ±√(13/4)$

$x + 2 = ±√13 / 2$

Finally, subtract 2 from both sides to solve for $x$:

$x = -2 ± √13 / 2$

Thus, the solutions for the quadratic equation are:

$x = (-4 + √13) / 2$ and $x = (-4 - √13) / 2$

Conclusion

In conclusion, the first step Maya should take when solving the quadratic equation $4x^2 + 16x + 3 = 0$ by completing the square is to isolate the constant term. This involves subtracting 3 from both sides of the equation, resulting in $4x^2 + 16x = -3$. This step is fundamental because it sets the stage for the subsequent steps, allowing for a clearer and more manageable process of completing the square. By understanding the logic behind each step, you can confidently apply this method to solve various quadratic equations.

We have also outlined the remaining steps in the process, from dividing by the coefficient of $x^2$ to finding the final solutions. Completing the square is a valuable technique in algebra, and mastering it will enhance your problem-solving skills. Remember, the first step is always to isolate the constant, paving the way for a successful solution. This comprehensive guide should provide you with a clear understanding of the initial step and the overall process, empowering you to tackle quadratic equations with confidence.