Completing The Square: Find The Missing Number

Hey guys! Today, we're diving into a classic algebra technique called "completing the square." It's a super handy method for solving quadratic equations, and it's all about transforming one side of the equation into a perfect square trinomial. But to do that, we need to figure out the magic number that makes it all work. So, let's break down the question: What number needs to be added to both sides of the equation $x^2 + 12x = 11$ to complete the square? We will explore everything you need to know.

Understanding Completing the Square

Before we jump into solving our specific equation, let's quickly recap the idea behind completing the square. At its heart, this technique is about manipulating a quadratic expression of the form $x^2 + bx$ into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. When you expand these, you get $x^2 + 2ax + a^2$ or $x^2 - 2ax + a^2$. The key thing to notice here is the relationship between the coefficient of the x term (that's the $2a$ or $-2a$) and the constant term (the $a^2$). To find the constant term needed to complete the square, we take half of the coefficient of the x term and then square it. This is the golden rule in completing the square, and it's what allows us to turn a quadratic expression into a perfect square trinomial. By understanding this relationship, we can strategically add a specific number to both sides of an equation, transforming it into a solvable form. This method not only simplifies solving quadratic equations but also provides a deeper insight into the structure and properties of quadratic expressions.

Identifying the Key Components

Okay, so let's bring it back to our equation: $x^2 + 12x = 11$. To figure out what number we need to add, we first need to identify the coefficient of our $x$ term. In this case, it's 12. Remember, the coefficient is simply the number that's multiplying the $x$. Now, here comes the crucial step: we take half of this coefficient. Half of 12 is 6. This is a critical intermediate value because it directly relates to the constant term we're aiming for. Next, we square this result. So, we square 6, which gives us 36. This number, 36, is the magic number we've been searching for. It's the value that, when added to the left side of the equation, will turn the expression $x^2 + 12x$ into a perfect square trinomial. By meticulously following these steps—identifying the coefficient, halving it, and then squaring the result—we pinpoint the exact value needed to complete the square. This process not only simplifies the equation but also paves the way for easy factorization and solution.

Applying the Magic Number

Now that we've found our magic number, 36, let's add it to both sides of the equation. Adding the same number to both sides keeps the equation balanced, which is super important in algebra. So, our equation now looks like this: $x^2 + 12x + 36 = 11 + 36$. Notice that we've added 36 to both the left side and the right side. This ensures that the equation remains equivalent to its original form while also setting us up to complete the square. On the left side, we now have the trinomial $x^2 + 12x + 36$. The beauty of this is that it's a perfect square trinomial, meaning it can be factored into a binomial squared. On the right side, we simply add 11 and 36 to get 47. So, our updated equation is $x^2 + 12x + 36 = 47$. By adding the magic number to both sides, we've transformed the equation into a form that's much easier to solve, bringing us closer to finding the values of x that satisfy the equation.

Factoring and Simplifying

Alright, so we've got $x^2 + 12x + 36 = 47$. Now comes the fun part: factoring the left side. Remember, we added 36 specifically to make the left side a perfect square trinomial. This means it can be factored into the form $(x + a)^2$. In our case, the trinomial $x^2 + 12x + 36$ factors neatly into $(x + 6)^2$. Why? Because 6 is half of 12 (the coefficient of our x term), and 6 squared is 36 (our constant term). So, we can rewrite our equation as $(x + 6)^2 = 47$. This is a significant step because we've transformed a quadratic equation into a simpler form where we have a squared term equal to a constant. This form is much easier to solve for x. By factoring the perfect square trinomial, we've essentially condensed the left side of the equation into a more manageable expression, paving the way for the final steps in solving for the variable.

Solving for x

We're in the home stretch now! We've got $(x + 6)^2 = 47$. To solve for $x$, we need to get rid of that square. The way we do that is by taking the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative roots. So, we get $x + 6 = ±\[sqrt{47]}$. Now, we simply subtract 6 from both sides to isolate $x$. This gives us $x = -6 ± \[sqrt{47}]$. This means we actually have two solutions: $x = -6 + \[sqrt{47}]$ and $x = -6 - \[sqrt{47}]$. These are the values of $x$ that make the original equation true. By taking the square root and isolating x, we've successfully found the solutions to the quadratic equation. This final step demonstrates the power of completing the square as a method for solving equations that might not be easily factorable by other means.

Conclusion

So, to answer the original question, the number that needs to be added to both sides of the equation $x^2 + 12x = 11$ to complete the square is 36. We found this by taking half of the coefficient of the x term (which was 12), squaring it (6 squared is 36), and then adding that number to both sides. Completing the square is a powerful technique, guys, and understanding the steps can help you solve a wide range of quadratic equations. Keep practicing, and you'll be a pro in no time! By mastering this method, you gain not only a valuable problem-solving tool but also a deeper understanding of the underlying principles of algebra. Good job, and keep up the great work!