Solving 36x^2 = 25: A Step-by-Step Factoring Guide
Hey guys! Today, we're going to dive into solving a quadratic equation using factoring. Specifically, we'll tackle the equation 36x^2 = 25. Don't worry, it's not as intimidating as it looks! We’ll break it down step-by-step so you can easily follow along. Factoring is a crucial skill in algebra, and mastering it can make solving more complex problems a breeze. So, let's get started and see how we can find the solutions to this equation together!
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations are super common in math and have a variety of real-world applications, from physics to engineering. The solutions to a quadratic equation are also known as its roots or zeros, which are the values of x that make the equation true.
In our case, the equation 36x^2 = 25 is a quadratic equation, but it's not in the standard form yet. To solve it by factoring, we first need to rearrange it into the ax^2 + bx + c = 0 format. This involves moving all terms to one side of the equation, leaving zero on the other side. It might seem like a simple step, but it's crucial for setting up the problem for factoring. Once we have the equation in the standard form, we can then look for factors that help us break down the equation and find the solutions. Understanding this basic structure is key to successfully solving quadratic equations.
Step 1: Rearrange the Equation
The first thing we need to do is get our equation, 36x^2 = 25, into the standard quadratic form, which, as we mentioned earlier, is ax^2 + bx + c = 0. To do this, we need to move the 25 from the right side of the equation to the left side. We can achieve this by subtracting 25 from both sides of the equation. This maintains the balance of the equation and ensures we're not changing the solutions.
So, let's subtract 25 from both sides:
36x^2 - 25 = 25 - 25
This simplifies to:
36x^2 - 25 = 0
Now our equation is in the standard quadratic form, where a = 36, b = 0 (since there's no x term), and c = -25. This form is essential because it allows us to use various factoring techniques effectively. Recognizing this form is the first major step in solving the equation. With the equation rearranged, we're now ready to move on to the next step: factoring!
Step 2: Factor the Difference of Squares
Now that we have our equation in the standard form, 36x^2 - 25 = 0, we can start factoring. Notice anything special about this equation? It's a difference of squares! The difference of squares is a common pattern in algebra where we have the form a^2 - b^2, which can be factored into (a - b)(a + b). Recognizing this pattern is super helpful because it simplifies the factoring process significantly.
In our case, 36x^2 is a perfect square, as it can be written as (6x)^2, and 25 is also a perfect square, as it's 5^2. So, we can apply the difference of squares pattern. Let's identify a and b in our equation:
- a = 6x
- b = 5
Now, we can factor 36x^2 - 25 using the formula (a - b)(a + b). Plugging in our values for a and b, we get:
(6x - 5)(6x + 5) = 0
Awesome! We've successfully factored the quadratic equation. This step is crucial because it breaks down the equation into a product of two binomials, which makes it much easier to find the solutions. With the equation factored, we're now ready to move on to the final step: solving for x.
Step 3: Solve for x
Okay, guys, we're in the home stretch! We've factored our equation into (6x - 5)(6x + 5) = 0. Now, we need to find the values of x that make this equation true. Here's where the zero-product property comes in handy. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if A * B = 0*, then either A = 0 or B = 0 (or both).
So, we can set each factor equal to zero and solve for x:
- 6x - 5 = 0
- 6x + 5 = 0
Let's solve each equation separately.
For the first equation, 6x - 5 = 0, we add 5 to both sides:
6x = 5
Then, we divide by 6:
x = 5/6
Now, let's solve the second equation, 6x + 5 = 0. We subtract 5 from both sides:
6x = -5
Then, we divide by 6:
x = -5/6
And there we have it! We've found our two solutions for x. These are the values that make the original equation, 36x^2 = 25, true. Great job!
Step 4: Verify the Solutions
Before we wrap things up, it's always a good idea to check our answers. Verifying the solutions ensures that we haven't made any mistakes along the way. To do this, we'll plug each value of x back into the original equation, 36x^2 = 25, and see if it holds true.
First, let's check x = 5/6:
36(5/6)^2 = 25
36(25/36) = 25
25 = 25
Great! The equation holds true for x = 5/6. Now, let's check x = -5/6:
36(-5/6)^2 = 25
36(25/36) = 25
25 = 25
Awesome! The equation also holds true for x = -5/6. This confirms that both of our solutions are correct. Verifying solutions is a crucial step in problem-solving because it gives us confidence in our answers and helps us catch any errors. We’ve done a thorough job by checking our work, so we can be sure we’ve got the correct solutions.
Final Answer
Alright, guys, let's bring it all together! We set out to solve the quadratic equation 36x^2 = 25 by factoring, and we've successfully done it. We rearranged the equation into standard form, factored it using the difference of squares, and then solved for x using the zero-product property. We even took the extra step to verify our solutions, ensuring we got everything right. So, what are our solutions?
The solutions to the quadratic equation 36x^2 = 25 are:
- x = 5/6
- x = -5/6
These are the values of x that make the equation true. We expressed our answers as reduced fractions, just as the problem asked. Factoring can be a super useful technique for solving quadratic equations, especially when you recognize patterns like the difference of squares. By breaking down the problem into smaller, manageable steps, we were able to find the solutions efficiently and accurately.
Conclusion
So, there you have it! We've successfully solved the quadratic equation 36x^2 = 25 by factoring. Remember, the key steps are to rearrange the equation into standard form, factor it appropriately (in this case, using the difference of squares), set each factor equal to zero, and solve for x. And don't forget to verify your solutions to ensure accuracy! Factoring quadratic equations is a fundamental skill in algebra, and the more you practice, the more comfortable you'll become with it. Keep up the great work, and you'll be solving all sorts of equations in no time!