Solving Quadratic Equations Finding Solutions For 4x^2 - 36 = 0

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Hey guys! Today, we are diving into the fascinating world of quadratic equations, and we're going to tackle a specific problem: finding the solutions for the equation 4x^2 - 36 = 0. If you've ever wondered how to solve these types of equations, you're in the right place. We’ll break it down step by step so that everyone can follow along. Let's jump right in and make math a little less mysterious and a lot more fun!

Understanding Quadratic Equations

First off, what exactly is a quadratic equation? Quadratic equations are polynomial equations of the second degree. This means they have the general form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we're trying to solve for. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These are the values of 'x' that make the equation true.

In our case, we have the equation 4x^2 - 36 = 0. Notice that it fits the general form, with a = 4, b = 0 (since there's no 'x' term), and c = -36. Now, let's explore the different methods we can use to find the solutions.

Methods for Solving Quadratic Equations

There are several ways to solve quadratic equations, and we'll briefly touch on a few before focusing on the method that's most efficient for our specific equation.

  1. Factoring: This method involves rewriting the quadratic equation as a product of two binomials. It's a great technique when the equation can be factored easily.
  2. Completing the Square: This method involves manipulating the equation to form a perfect square trinomial. It's a bit more involved but works for any quadratic equation.
  3. Quadratic Formula: This is the most general method and can be used to solve any quadratic equation. The quadratic formula is given by: x = [-b ± √(b^2 - 4ac)] / (2a).
  4. Isolating the Variable: For some quadratic equations, like the one we're dealing with today, we can isolate the variable directly. This is often the quickest and easiest method when applicable.

Given our equation 4x^2 - 36 = 0, isolating the variable seems like the most straightforward approach. Let’s dive into that method now!

Solving 4x^2 - 36 = 0 by Isolating the Variable

Okay, guys, let’s get our hands dirty and solve this equation step by step. Isolating the variable means we want to get 'x' by itself on one side of the equation. Here’s how we do it:

Step 1: Add 36 to both sides

We start with our original equation: 4x^2 - 36 = 0. To isolate the term with 'x', we add 36 to both sides of the equation: 4x^2 - 36 + 36 = 0 + 36 This simplifies to: 4x^2 = 36

Step 2: Divide both sides by 4

Now, we want to get rid of the coefficient (the number in front of) x^2, which is 4. To do this, we divide both sides of the equation by 4: (4x^2) / 4 = 36 / 4 This simplifies to: x^2 = 9

Step 3: Take the square root of both sides

We're almost there! We now have x^2 = 9. To find 'x', we need to take 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 because both (3)^2 and (-3)^2 equal 9. So, we have: √x^2 = ±√9 This gives us: x = ±3

Step 4: Identify the solutions

Therefore, the solutions to the equation 4x^2 - 36 = 0 are x = 3 and x = -3. We've found our roots! Easy peasy, right?

Verifying the Solutions

It's always a good idea to check our answers to make sure they’re correct. Let's plug our solutions back into the original equation to verify.

Check x = 3: 4(3)^2 - 36 = 4(9) - 36 = 36 - 36 = 0 So, x = 3 is indeed a solution.

Check x = -3: 4(-3)^2 - 36 = 4(9) - 36 = 36 - 36 = 0 And x = -3 is also a solution.

Both solutions check out! We can be confident in our answer.

Why Understanding Solutions Matters

Finding the solutions to equations like this isn't just an exercise in algebra; it has real-world applications. Quadratic equations pop up in physics, engineering, economics, and many other fields. For example, they can be used to model the trajectory of a projectile, the shape of a suspension bridge, or the growth of a population. Understanding how to solve quadratic equations opens doors to solving a wide range of problems in the real world.

Imagine you’re designing a bridge, and you need to calculate the curve of the supporting cables. Quadratic equations can help you determine the exact shape needed to ensure the bridge is stable. Or, if you’re a physicist trying to predict how far a ball will travel when thrown, you’ll use quadratic equations to model its path. The possibilities are endless!

Common Mistakes to Avoid

When solving quadratic equations, there are a few common pitfalls you should try to avoid. Let's run through a couple of these so you can steer clear of them.

  1. Forgetting the Negative Root: As we saw in our example, when taking the square root, it’s crucial to remember both the positive and negative roots. Forgetting the negative root is a common mistake that can lead to missing a solution.
  2. Incorrectly Applying the Order of Operations: When plugging solutions back into the equation to verify, make sure to follow the order of operations (PEMDAS/BODMAS). This means doing exponents before multiplication and addition/subtraction.
  3. Making Arithmetic Errors: Simple arithmetic errors can throw off your entire solution. Double-check your calculations at each step to minimize mistakes.

By being mindful of these common errors, you'll increase your accuracy and confidence in solving quadratic equations.

Practice Problems

Now that we’ve worked through an example together, it’s time for you to try your hand at solving some quadratic equations. Practice makes perfect, guys! Here are a couple of problems for you to tackle:

  1. Solve 2x^2 - 50 = 0
  2. Find the solutions to 3x^2 - 27 = 0

Work through these problems using the method we discussed, and feel free to check your answers by plugging them back into the original equations. If you get stuck, don’t worry! Go back through the steps we covered, and remember, the key is to isolate the variable.

Exploring Other Solution Methods

While isolating the variable was the most efficient method for our equation, let's briefly explore how we could have used other methods to solve 4x^2 - 36 = 0.

Factoring

We could factor the left side of the equation as a difference of squares. Recall that a^2 - b^2 = (a - b)(a + b). In our case, 4x^2 - 36 can be rewritten as (2x)^2 - (6)^2. So, we can factor the equation as follows: 4x^2 - 36 = (2x - 6)(2x + 6) = 0 To find the solutions, we set each factor equal to zero: 2x - 6 = 0 or 2x + 6 = 0 Solving these linear equations gives us: x = 3 or x = -3

Quadratic Formula

We could also use the quadratic formula, although it's a bit overkill for this particular equation. Remember, the quadratic formula is: x = [-b ± √(b^2 - 4ac)] / (2a) For our equation 4x^2 - 36 = 0, we have a = 4, b = 0, and c = -36. Plugging these values into the formula, we get: x = [-0 ± √(0^2 - 4(4)(-36))] / (2(4)) x = [± √(576)] / 8 x = [± 24] / 8 This simplifies to: x = 3 or x = -3

As you can see, both factoring and the quadratic formula lead us to the same solutions, but isolating the variable was the most direct method in this case.

Conclusion

Alright, guys, we've covered a lot today! We successfully solved the quadratic equation 4x^2 - 36 = 0 by isolating the variable, and we verified our solutions. We also touched on other methods, like factoring and the quadratic formula, and discussed common mistakes to avoid. Remember, understanding quadratic equations is a valuable skill that has applications in various fields.

So, keep practicing, keep exploring, and don't be afraid to tackle those math problems head-on. You've got this! And remember, if you ever feel stuck, just break the problem down into smaller steps and take it one step at a time. You’ll get there. Until next time, happy solving!