Hey guys! Today, we're diving into the world of quadratic equations, specifically focusing on how to solve the equation (x-7)^2 = 36. This is a common type of problem you'll encounter in algebra, and mastering it can really boost your math skills. So, let's break it down step by step and make sure we understand every little detail.
Understanding Quadratic Equations
Before we jump into solving our specific equation, it's super helpful to understand what a quadratic equation actually is. At its core, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Quadratic equations pop up in all sorts of real-world scenarios, from physics to engineering to even economics, so getting comfy with them is a smart move. They often describe parabolic curves, which you might recognize from graphs and diagrams. Understanding the nature of these equations helps in tackling more complex problems later on. The solutions to a quadratic equation are also known as its roots or zeros, and these are the values of x that make the equation true. Finding these values is what we're aiming for when we solve a quadratic equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages and is suited to different types of equations. For our problem, we'll use a method that takes advantage of the specific form of the equation, making it straightforward to solve. Now that we have a solid grasp of the basics, let's move on to our equation and start finding those values of x.
Method 1: Solving by Taking the Square Root
The equation we're tackling today is (x-7)^2 = 36. This equation is set up perfectly for a method called "taking the square root." This method is super efficient when you have a squared term isolated on one side of the equation, like we do here. The basic idea is to undo the square by taking the square root of both sides. But, and this is a big but, we need to remember that when we take the square root of a number, we get both a positive and a negative solution. So, let's get started.
Step 1: Take the Square Root of Both Sides
First things first, we take the square root of both sides of the equation. This gives us: √((x-7)^2) = ±√(36). Notice that we've included the ± (plus or minus) sign on the right side. This is crucial because both 6 and -6, when squared, give us 36. So, we have to consider both possibilities. This step essentially unwraps the squared term on the left side, making the equation much easier to handle. By remembering to include both positive and negative roots, we ensure that we capture all possible solutions for x. This is a common pitfall for many students, so it's worth highlighting and remembering. Taking the square root is a powerful tool for solving equations, especially when dealing with perfect squares like in our case. The simplicity of this step sets us up nicely for the next phase of our solution.
Step 2: Simplify the Equation
After taking the square root, our equation looks like this: x - 7 = ±6. This is much simpler than where we started, right? Now, we have two separate equations to solve: x - 7 = 6 and x - 7 = -6. This split is necessary because we need to account for both the positive and negative square roots of 36. Simplifying the equation in this way makes it clear that we have two distinct paths to follow to find our solutions. Each of these equations is a basic linear equation, which we can solve with a simple addition. The clarity gained in this step helps prevent errors and ensures that we don't miss any potential solutions. By breaking down the equation into two separate cases, we make the problem more manageable and set ourselves up for a straightforward final step.
Step 3: Solve for x
Now, let's solve each of these equations individually. For x - 7 = 6, we add 7 to both sides to isolate x, which gives us x = 6 + 7, so x = 13. That's one solution down! For x - 7 = -6, we do the same thing: add 7 to both sides. This gives us x = -6 + 7, so x = 1. And there's our second solution! So, the values of x that satisfy the original equation are 13 and 1. These solutions are the points where the parabola represented by the equation intersects the x-axis. We've successfully navigated through the equation, remembering to account for both positive and negative roots along the way. This careful approach ensures we haven't missed any solutions. Solving for x in this manner is a fundamental skill in algebra, and mastering it will greatly aid in tackling more complex problems.
The Solutions
So, after walking through the steps, we've found that the values of x that satisfy the equation (x-7)^2 = 36 are x = 13 and x = 1. Looking back at our multiple-choice options:
- A. x = 13 (Correct!)
- B. x = 1 (Correct!)
- C. x = -29 (Incorrect)
- D. x = 42 (Incorrect)
We can confidently select options A and B as the correct answers. This reinforces the importance of methodical problem-solving, especially in math. By breaking down the problem into manageable steps and carefully executing each one, we've arrived at the correct solutions. Verifying our answers against the original equation is a great way to ensure accuracy and build confidence in our problem-solving abilities. Now, let's zoom out and discuss another method for solving this type of equation.
Method 2: Expanding and Factoring (Alternative Method)
While taking the square root is super efficient for this particular equation, it's always good to have other methods in your toolkit. Another way to solve (x-7)^2 = 36 is by expanding the squared term and then factoring. This method is a bit more involved, but it's a valuable technique for other quadratic equations that might not be as easily solved by taking the square root.
Step 1: Expand the Squared Term
First, we need to expand (x-7)^2. Remember, this means (x-7)(x-7). We can use the FOIL method (First, Outer, Inner, Last) to expand this:
- First: x * x = x^2
- Outer: x * -7 = -7x
- Inner: -7 * x = -7x
- Last: -7 * -7 = 49
Combining these, we get x^2 - 7x - 7x + 49, which simplifies to x^2 - 14x + 49. Expanding the squared term transforms the equation into a standard quadratic form, which we can then manipulate further. This step is crucial for applying methods like factoring or the quadratic formula. By expanding, we reveal the underlying structure of the equation and make it amenable to these other solution techniques. The expanded form also allows us to see all the terms and coefficients clearly, which is helpful in identifying the best approach for solving the equation.
Step 2: Rewrite the Equation in Standard Form
Now, our equation looks like this: x^2 - 14x + 49 = 36. To use factoring, we need to set the equation equal to zero. So, we subtract 36 from both sides: x^2 - 14x + 49 - 36 = 0. This simplifies to x^2 - 14x + 13 = 0. Rewriting the equation in standard form is a key step in solving quadratic equations by factoring or using the quadratic formula. It aligns the equation with the general form ax^2 + bx + c = 0, making it easier to identify the coefficients and apply the appropriate methods. This step ensures that we can easily factor the quadratic expression or plug the values into the quadratic formula without confusion. By setting the equation to zero, we create a clear target for factoring, which is to find two binomials that multiply to give the quadratic expression.
Step 3: Factor the Quadratic Expression
Next, we need to factor the quadratic expression x^2 - 14x + 13. We're looking for two numbers that multiply to 13 and add up to -14. Those numbers are -13 and -1. So, we can factor the expression as (x - 13)(x - 1) = 0. Factoring is a powerful technique for solving quadratic equations, as it allows us to break down the expression into simpler components. The factored form directly reveals the solutions of the equation, as each factor corresponds to a root. By finding the correct factors, we essentially reverse the process of expansion and rewrite the equation in a form that makes the solutions immediately apparent. This step requires a good understanding of number properties and the relationships between the coefficients and the factors.
Step 4: Solve for x
Now that we have (x - 13)(x - 1) = 0, we can use the zero-product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, either x - 13 = 0 or x - 1 = 0. Solving these gives us x = 13 and x = 1. And guess what? These are the same solutions we found using the square root method! Applying the zero-product property is the final step in solving a quadratic equation by factoring. It transforms the factored equation into two simpler linear equations, each of which can be easily solved for x. This property is fundamental to the factoring method and provides a direct link between the factors of the quadratic expression and the solutions of the equation. By setting each factor equal to zero, we isolate the possible values of x that make the equation true, leading us to the solutions.
Conclusion
Alright, guys, we've tackled the equation (x-7)^2 = 36 using two different methods: taking the square root and expanding and factoring. Both methods led us to the same solutions: x = 13 and x = 1. This highlights the beauty of math – there are often multiple paths to the correct answer! The key takeaway here is to choose the method that feels most comfortable and efficient for you, but also to be familiar with other methods in case you encounter a problem where one approach is clearly superior. Understanding various solution techniques expands your problem-solving toolkit and makes you a more versatile mathematician. Whether it's taking the square root, factoring, or using the quadratic formula, each method offers unique advantages and can be applied in different contexts. Practice is crucial in mastering these techniques and developing the intuition to choose the best approach for any given problem. Keep practicing, and you'll become a quadratic equation-solving pro in no time!
So, the next time you see an equation like this, you'll know exactly what to do. Keep practicing, and happy solving!