Solving Quadratic Equations By Completing The Square Step-by-Step

Hey guys! Today, we're diving into the exciting world of quadratic equations and tackling a common method for solving them: completing the square. We'll break down the process step-by-step, making sure you've got a solid understanding of how it works. Let's jump right in!

Understanding the Problem

Before we dive into the solution, let's make sure we fully understand the question. We're given the first few steps of solving the quadratic equation 8x^2 + 80x = -5 by completing the square. The initial steps are shown, and our goal is to figure out which number is missing in the last step:

8x^2 + 80x = -5
8(x^2 + 10x) = -5
8(x^2 + 10x + 25) = -5 + ______

Our mission, should we choose to accept it (and we do!), is to find that missing number. To do this, we'll need to understand the process of completing the square.

What is Completing the Square?

Completing the square is a technique used to rewrite a quadratic equation in a form that allows us to easily solve for the variable (in this case, 'x'). The goal is to transform the quadratic expression into a perfect square trinomial, which can then be factored into the square of a binomial. This method is super useful when the quadratic equation doesn't factor easily or when we want to rewrite the equation in vertex form.

The general idea behind completing the square is to manipulate a quadratic expression of the form ax^2 + bx + c into the form a(x + h)^2 + k, where h and k are constants. This form makes it straightforward to find the vertex of the parabola represented by the quadratic equation, and it also simplifies solving for the roots (the values of x that make the equation equal to zero).

Now, let's dive into the specific steps involved in completing the square, using our example problem as a guide. This will help us not only solve the problem at hand but also equip you with the skills to tackle similar problems in the future. Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become with the process. We'll break it down into manageable chunks, so you can follow along easily. Let's get started!

Step-by-Step Solution

Let's walk through the steps of completing the square, focusing on our given equation: 8x^2 + 80x = -5. This will help us pinpoint the missing number in the final step.

Step 1: Factor out the Leading Coefficient

The first step is to factor out the leading coefficient (the coefficient of the x^2 term) from the terms containing x. In our case, the leading coefficient is 8. Factoring 8 from the left side of the equation 8x^2 + 80x = -5, we get:

8(x^2 + 10x) = -5

This step is crucial because it sets us up to create a perfect square trinomial inside the parentheses. By factoring out the leading coefficient, we ensure that the coefficient of the x^2 term inside the parentheses is 1, which is a necessary condition for completing the square. Now, we have a simpler quadratic expression inside the parentheses that we can work with. This makes the subsequent steps much easier to manage and reduces the chances of making errors. So, factoring out the leading coefficient is a key initial step in the process.

Step 2: Complete the Square

Now, we focus on the expression inside the parentheses: x^2 + 10x. To complete the square, we need to add a constant term that will make this expression a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial (e.g., (x + a)^2). The constant term we need to add is determined by taking half of the coefficient of the x term, squaring it.

In our expression, the coefficient of the x term is 10. Half of 10 is 5, and squaring 5 gives us 25. So, we need to add 25 inside the parentheses to complete the square:

8(x^2 + 10x + 25)

Now, x^2 + 10x + 25 is a perfect square trinomial because it can be factored as (x + 5)^2. This is the magic of completing the square – we've transformed a quadratic expression into a form that's much easier to work with. But, and this is a crucial but, we can't just add 25 inside the parentheses without making a corresponding adjustment to the other side of the equation. This is where the next step comes in.

Step 3: Balance the Equation

Here's where it gets a little tricky, but stay with me! We've added 25 inside the parentheses, but remember that the entire expression inside the parentheses is being multiplied by 8. So, we're not just adding 25 to the left side of the equation; we're actually adding 8 * 25.

8 multiplied by 25 equals 200. To maintain the balance of the equation, we must add 200 to the right side as well. This gives us:

8(x^2 + 10x + 25) = -5 + 200

This is the crucial step where we find the missing number! We've added 200 to the right side to balance the equation. This ensures that the equation remains equivalent to the original equation, even after we've manipulated it by completing the square. Failing to balance the equation would lead to an incorrect solution, so this step is absolutely essential. Now, we're one step closer to identifying the missing number and solving the quadratic equation.

The Missing Number

Looking back at the last step:

8(x^2 + 10x + 25) = -5 + ______

We can now clearly see that the missing number is 200. This is because we added 8 * 25 = 200 to the left side of the equation when we completed the square, and we needed to add the same amount to the right side to maintain balance. Therefore, the correct answer is 200. We've successfully identified the missing number by carefully following the steps of completing the square and paying close attention to the need for balancing the equation. This process not only answers the specific question but also reinforces the understanding of the fundamental principles behind completing the square.

So, the answer is 200.

Why the Other Options Are Incorrect

It's helpful to understand why the other options are incorrect to solidify our understanding of the process. Let's briefly discuss why the other choices aren't the right fit:

• -200: This is incorrect because we need to add a positive value to balance the equation. We added a positive value (25) inside the parentheses, which was then multiplied by 8, resulting in a positive addition to the left side. Therefore, we need to add a positive value to the right side as well.

Understanding why the incorrect options are wrong is just as important as knowing why the correct answer is right. It helps to deepen our comprehension of the concepts and avoid making similar mistakes in the future. By analyzing the errors, we can reinforce the correct steps and reasoning, leading to a more solid grasp of the material. So, let's keep this in mind as we continue to practice and learn.

Next Steps in Solving the Equation

While we've found the missing number, let's briefly discuss the next steps in solving the quadratic equation. This will give you a complete picture of the process.

Step 4: Factor and Simplify

First, we factor the perfect square trinomial and simplify the right side:

8(x + 5)^2 = 195

Step 5: Isolate the Squared Term

Next, we divide both sides by 8:

(x + 5)^2 = 195/8

Step 6: Take the Square Root

Now, take the square root of both sides (remembering both positive and negative roots):

x + 5 = ±√(195/8)

Step 7: Solve for x

Finally, subtract 5 from both sides to solve for x:

x = -5 ± √(195/8)

These are the solutions to the quadratic equation. Completing the square allows us to find these solutions even when the equation doesn't factor easily. By understanding each step and the reasoning behind it, you can confidently tackle a wide range of quadratic equations. Remember, the key is to practice and break down the problem into manageable steps. Keep up the great work!

Key Takeaways

• Completing the square is a powerful technique for solving quadratic equations.
• Remember to balance the equation by adding the same value to both sides.
• The missing number in this problem was 200.

Practice Makes Perfect

The best way to master completing the square is to practice! Work through various examples, and don't be afraid to make mistakes – they're part of the learning process. With each problem you solve, you'll become more confident and proficient in this valuable technique. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Remember, every expert was once a beginner, so keep at it, and you'll see your skills grow.

So, there you have it! We've successfully navigated the process of completing the square, identified the missing number, and discussed the next steps in solving the quadratic equation. Remember, mathematics is a journey, not a destination, so enjoy the process of learning and discovery. Keep challenging yourself, keep practicing, and keep exploring the fascinating world of mathematics. You've got this!