Solving Quadratic Equations: Finding Positive Solutions

Hey math enthusiasts! Let's dive into a classic problem: "The difference of the square of a number and 4 is equal to 3 times that number. Find the positive solution." This isn't just some random math puzzle; it's a fantastic example of how we use quadratic equations in the real world. We'll break it down step by step, making sure it's crystal clear. So, grab your pencils, and let's get started!

Understanding the Problem: Translating Words into Math

Alright, guys, the first step in tackling any math problem is to understand what it's asking. In this case, we have a sentence that's practically screaming for us to turn it into an equation. Let's break it down piece by piece. The core of the problem lies in the relationship between a number, its square, and the constant value 4, with 3 times of that number. We can start by stating that the difference of the square of a number and 4 can be represented by x^2 - 4. Also, it's mentioned that is equal to 3 times that number can be represented by 3x. Therefore, the phrase can be converted into the equation format. This is the heart of the problem. Translating words into mathematical symbols is the key to solving it. We know our equation is going to involve a square (x squared), a regular 'x', and some numbers. Remember that the ultimate goal is to find the value of x that makes the equation true. Before we begin, it's important to remember the basics of quadratic equations. They take the form ax^2 + bx + c = 0, where a, b, and c are constants. The values of a, b, and c determine the shape and position of the parabola. Solving the equation means finding the roots of the equation, which are the points where the parabola intersects the x-axis. These roots are also known as solutions to the quadratic equation. So we are not only going to find a positive solution, but we are also going to find other solutions if possible.

Here’s how we can translate the problem statement into a mathematical equation:

• "The difference of the square of a number and 4": This translates to x^2 - 4 (where 'x' is our unknown number).
• "is equal to 3 times that number": This becomes = 3x.

Putting it all together, we get our equation: x^2 - 4 = 3x.

Now, before we jump into solving this, it's a good idea to rearrange the equation into the standard quadratic form (ax^2 + bx + c = 0). This helps us apply standard solving methods more easily. Moving the 3x to the left side, we get x^2 - 3x - 4 = 0. See? Now it looks like a proper quadratic equation! So this problem gives us a wonderful opportunity to hone our skills in algebra and become more familiar with quadratic equations. We’ll cover the main methods for solving quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method has its pros and cons, and knowing which one to use at the right time can save a lot of effort and ensure you get the right answer quickly. In this problem, after we have our equation in the standard form, we want to figure out which approach is the best for solving it. Usually, factoring is the simplest method if it’s possible. However, not all quadratic equations are easily factorable, and that is where the quadratic formula comes to the rescue. Completing the square is another approach, which is useful when working with quadratic equations, but it is sometimes more involved than other methods.

Solving the Equation: Methods and Solutions

Okay, team, now that we have our equation x^2 - 3x - 4 = 0, it's time to find the value(s) of x that make it true. There are several ways to solve a quadratic equation like this. Let's look at a few of the most common ones.

Factoring Method

This is often the quickest method if it works. Factoring means breaking down the quadratic expression into two simpler expressions that multiply together to give the original expression. For our equation, we're looking for two numbers that multiply to -4 (the constant term) and add up to -3 (the coefficient of the x term). Those numbers are -4 and 1. So we can factor the equation as follows: (x - 4)(x + 1) = 0. From here, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions:

• x - 4 = 0 => x = 4
• x + 1 = 0 => x = -1

Quadratic Formula Method

If factoring doesn't work (or if you just prefer a more general approach), you can always use the quadratic formula. This formula works for any quadratic equation in the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 1, b = -3, and c = -4. Plugging these values into the formula, we get:

x = (3 ± √((-3)^2 - 4 * 1 * -4)) / (2 * 1) x = (3 ± √(9 + 16)) / 2 x = (3 ± √25) / 2 x = (3 ± 5) / 2

This gives us two solutions:

• x = (3 + 5) / 2 = 8 / 2 = 4
• x = (3 - 5) / 2 = -2 / 2 = -1

Notice that we got the same solutions as with the factoring method. Whether you factor or use the quadratic formula, the solutions are x = 4 and x = -1.

We need to find the positive solution. Therefore, the positive solution is x = 4.

Checking the Solution: Verification is Key

Always, and I mean always, check your answers! It's a great habit that helps you catch any silly mistakes. Let's plug our solutions (4 and -1) back into the original equation (x^2 - 4 = 3x) to make sure they work.

• For x = 4:

  • 4^2 - 4 = 16 - 4 = 12
  • 3 * 4 = 12
  • Since 12 = 12, x = 4 is a valid solution.

• For x = -1:

  • (-1)^2 - 4 = 1 - 4 = -3
  • 3 * -1 = -3
  • Since -3 = -3, x = -1 is also a valid solution.

Both of our solutions are correct! This step is super important because it confirms our understanding of the problem and our ability to solve it.

Conclusion: Wrapping Things Up

And there you have it, folks! We've successfully solved the problem and found the positive solution. The key takeaways from this problem are:

1. Translating words into math: This is a fundamental skill in algebra.
2. Rearranging equations: Putting equations in standard form makes them easier to solve.
3. Knowing multiple solution methods: Factoring and the quadratic formula are powerful tools.
4. Always checking your answer: This ensures accuracy and reinforces understanding.

So next time you encounter a word problem, remember these steps. Break down the problem, translate it into an equation, solve it using the appropriate method, and always check your work. Keep practicing, and you'll become a quadratic equation master in no time!

This entire process is just one of many great examples. From understanding the problem to providing a step-by-step approach, we were able to find the solution. Hopefully, this problem also helps you to understand the power of quadratic equations and how they can be used to solve different kinds of mathematical problems. Remember that the beauty of math is in its ability to model real-world scenarios, and problems like this are a perfect illustration of that. Keep practicing, and you will get better. Thanks for joining me today. Keep learning!