Solving X^2 + 20x + 100 = 36 A Step-by-Step Guide
Hey guys! Today, we're diving into a classic algebra problem: solving for x in the equation x² + 20x + 100 = 36. This might look a bit intimidating at first, but don't worry, we'll break it down step by step so you can conquer it like a math pro. We will explore different methods to tackle this equation, ensuring you understand the underlying concepts and can confidently solve similar problems in the future. So, grab your pencils and let's get started!
Understanding the Equation: x² + 20x + 100 = 36
Before we jump into solving, let's take a closer look at our equation: x² + 20x + 100 = 36. This is a quadratic equation, which means it has a term with x raised to the power of 2 (x²). Quadratic equations often have two solutions, and our goal is to find both values of x that make the equation true. This particular equation has a special form on the left-hand side. Notice that the expression x² + 20x + 100 is a perfect square trinomial. This is because it can be factored into (x + 10)². Recognizing this pattern is a huge shortcut to solving the problem. Perfect square trinomials are quadratic expressions that result from squaring a binomial. They follow the pattern (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b². In our case, a = x and b = 10, so x² + 2(x)(10) + 10² perfectly fits the pattern. This recognition allows us to rewrite the equation in a much simpler form, making it easier to solve. Ignoring this pattern and attempting to use other methods, while still possible, would likely involve more steps and a higher chance of making an error. Therefore, understanding the structure of the equation is critical for efficient and accurate problem-solving. So, keep your eyes peeled for these perfect square trinomials – they're your friends in the math world!
Method 1: Factoring and the Square Root Property
The most efficient way to solve this equation is by factoring and using the square root property. Remember how we identified that x² + 20x + 100 is a perfect square trinomial? That's the key here! We can rewrite the left side of the equation as (x + 10)²:
(x + 10)² = 36
Now, the square root property comes into play. This property states that if a² = b, then a = ±√b. In simpler terms, if something squared equals a number, then that something equals both the positive and negative square root of that number. Applying this to our equation, we take the square root of both sides:
√[(x + 10)²] = ±√36
This simplifies to:
x + 10 = ±6
Now we have two separate equations to solve:
- x + 10 = 6
- x + 10 = -6
Solving the first equation, we subtract 10 from both sides:
x = 6 - 10
x = -4
Solving the second equation, we also subtract 10 from both sides:
x = -6 - 10
x = -16
So, our two solutions are x = -4 and x = -16. We have successfully solved the equation by recognizing the perfect square trinomial, factoring, and applying the square root property. This method is often the quickest and cleanest way to solve quadratic equations that can be expressed as perfect squares. Remember, practice makes perfect! The more you encounter these types of problems, the faster you'll become at recognizing patterns and applying the appropriate techniques. Keep up the great work, and you'll be a factoring master in no time!
Method 2: Using the Quadratic Formula
Okay, so maybe you didn't spot the perfect square trinomial right away. No sweat! There's another trusty tool in our algebraic arsenal: the quadratic formula. This formula is a universal solution for any quadratic equation in the standard form of ax² + bx + c = 0. Our equation, x² + 20x + 100 = 36, isn't quite in this form yet, so let's rearrange it first. To get it into standard form, we need to subtract 36 from both sides:
x² + 20x + 100 - 36 = 0
This simplifies to:
x² + 20x + 64 = 0
Now we're in business! We can identify our coefficients: a = 1, b = 20, and c = 64. The quadratic formula itself looks like this:
x = (-b ± √(b² - 4ac)) / (2a)
It might look a bit intimidating, but it's just a matter of plugging in our values and simplifying. Let's do it! Substituting our values, we get:
x = (-20 ± √(20² - 4 * 1 * 64)) / (2 * 1)
Now, let's simplify step by step. First, calculate the value inside the square root:
20² - 4 * 1 * 64 = 400 - 256 = 144
So, our equation becomes:
x = (-20 ± √144) / 2
The square root of 144 is 12, so we have:
x = (-20 ± 12) / 2
Now we have two possible solutions, one with the plus sign and one with the minus sign:
- x = (-20 + 12) / 2
- x = (-20 - 12) / 2
Let's solve the first one:
x = -8 / 2
x = -4
And now the second one:
x = -32 / 2
x = -16
Guess what? We got the same solutions as before: x = -4 and x = -16. The quadratic formula might seem like a longer route in this case, but it's a reliable method that works for any quadratic equation, even those that don't factor easily. It's a powerful tool to have in your math toolkit!
Method 3: Completing the Square
Alright, let's explore another method for solving our equation: completing the square. This technique is particularly useful because it not only helps us solve quadratic equations, but it also provides insights into the structure of the equation itself. It's a bit more involved than factoring directly, but it's a valuable skill to master. Again, we start with our equation in its original form:
x² + 20x + 100 = 36
The first step in completing the square is to make sure the coefficient of the x² term is 1, which it already is in our case. Next, we focus on the x² and x terms (x² + 20x). We want to manipulate these terms to create a perfect square trinomial. To do this, we take half of the coefficient of our x term (which is 20), square it, and add it to both sides of the equation. Half of 20 is 10, and 10 squared is 100. But wait! We already have 100 on the left side of the equation. This means our equation is already set up perfectly for completing the square! This makes our job easier, as we can skip the addition step in this particular instance.
Now, we can rewrite the left side as a squared binomial, just like we did in Method 1:
(x + 10)² = 36
From this point on, the process is identical to what we did in Method 1. We take the square root of both sides:
√[(x + 10)²] = ±√36
Which simplifies to:
x + 10 = ±6
And then we solve for x as before:
- x + 10 = 6 => x = -4
- x + 10 = -6 => x = -16
So, once again, we arrive at the solutions x = -4 and x = -16. Completing the square might seem redundant in this example, given that we already had a perfect square trinomial, but it demonstrates the general principle of the method. In situations where the original equation doesn't readily present a perfect square, completing the square is a powerful technique for transforming the equation into a solvable form. It's a bit like having a Swiss Army knife for quadratic equations – it might not always be the fastest tool, but it's incredibly versatile!
Verifying the Solutions
We've solved for x using three different methods, and we've arrived at the same solutions each time: x = -4 and x = -16. But how can we be absolutely sure these are the correct answers? The best way is to verify our solutions by plugging them back into the original equation. This is a crucial step in problem-solving, as it helps us catch any potential errors we might have made along the way. Let's start with x = -4. We substitute -4 for x in the original equation:
(-4)² + 20(-4) + 100 = 36
Now, let's simplify:
16 - 80 + 100 = 36
36 = 36
The equation holds true! This confirms that x = -4 is indeed a valid solution. Now, let's do the same for x = -16:
(-16)² + 20(-16) + 100 = 36
Simplifying:
256 - 320 + 100 = 36
36 = 36
Again, the equation holds true! This confirms that x = -16 is also a valid solution. By verifying our solutions, we can be confident that we've solved the equation correctly. This practice also reinforces our understanding of what it means to
