Solving 7y² - 7y = 0: A Quadratic Formula Guide

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Hey guys! Let's dive into solving a quadratic equation today. We're going to tackle the equation 7y² - 7y = 0 using the quadratic formula. It might sound intimidating, but trust me, it's totally manageable. We'll break it down step by step so you can follow along easily. Whether you're brushing up on your algebra skills or tackling homework, this guide will help you master this type of problem. Let's get started and make math a little less mysterious!

Understanding the Quadratic Formula

Before we jump into the specific equation, let’s quickly recap what the quadratic formula is all about. The quadratic formula is a powerful tool for finding the solutions (also called roots or zeros) of any quadratic equation. Remember, a quadratic equation is an equation that can be written in the standard form:

ax² + bx + c = 0

Where a, b, and c are coefficients, and x is the variable we want to solve for. The quadratic formula itself is:

x = [-b ± √(b² - 4ac)] / 2a

This formula might look a bit scary at first, but it's really just a plug-and-chug situation once you identify your a, b, and c values. The ± symbol means we'll actually get two solutions: one where we add the square root part and one where we subtract it. These solutions are the points where the parabola (the graph of the quadratic equation) intersects the x-axis. In some cases, these solutions might be real numbers, and in other cases, they might be complex numbers. Understanding the quadratic formula is essential because it provides a universal method for solving any quadratic equation, regardless of whether it can be easily factored or not. Factoring is a great technique when it works, but the quadratic formula is the reliable workhorse that always gets the job done. So, keep this formula handy, and let's see how it works in practice with our equation!

Identifying Coefficients in 7y² - 7y = 0

Okay, now that we've refreshed our memory on the quadratic formula, let’s get to our equation: 7y² - 7y = 0. The first thing we need to do is identify the coefficients a, b, and c. This is super important because these values are what we're going to plug into the formula. Remember the standard form of a quadratic equation: ax² + bx + c = 0. In our case, the variable is y instead of x, but the principle is exactly the same.

Let's break it down:

  • The coefficient of the y² term (a) is 7. So, a = 7.
  • The coefficient of the y term (b) is -7. Don't forget that negative sign! So, b = -7.
  • Now, what about c? Notice that there's no constant term in our equation. That means c is simply 0. So, c = 0.

Identifying these coefficients correctly is the most critical step. A small mistake here will throw off the entire solution. So, always double-check your values before moving on. Once you've got a, b, and c, you're more than halfway there. The hard part is done! Now we're ready to plug these values into the quadratic formula and see what we get. Think of it like a recipe – you've got your ingredients, and now it's time to bake!

Applying the Quadratic Formula

Alright, we've got our coefficients: a = 7, b = -7, and c = 0. Now comes the fun part – plugging these values into the quadratic formula! Remember the formula?

y = [-b ± √(b² - 4ac)] / 2a

Let's substitute our values step by step. First, we'll replace a, b, and c with their corresponding numbers:

y = [-(-7) ± √((-7)² - 4 * 7 * 0)] / (2 * 7)

Notice how I've carefully put parentheses around the negative numbers. This is a good habit to avoid sign errors. Now, let's simplify this step-by-step:

  • -(-7) becomes +7, so we have y = [7 ± √((-7)² - 4 * 7 * 0)] / (2 * 7)
  • Next, let's deal with the terms inside the square root: (-7)² is 49, and 4 * 7 * 0 is 0. So we get y = [7 ± √(49 - 0)] / (2 * 7)
  • This simplifies to y = [7 ± √49] / (2 * 7)
  • The square root of 49 is 7, so we have y = [7 ± 7] / (2 * 7)
  • Finally, 2 * 7 is 14, giving us y = [7 ± 7] / 14

See? We've taken a seemingly complicated formula and broken it down into manageable pieces. Now we have a much simpler expression to work with. The ± sign means we have two paths to follow, one with addition and one with subtraction. Let's explore those next!

Calculating the Two Solutions

Okay, we've reached the point where we have y = [7 ± 7] / 14. Remember that ± sign means we actually have two separate calculations to do – one with addition and one with subtraction. This will give us our two solutions for y. Let's tackle them one at a time.

First, let's consider the addition case:

y₁ = (7 + 7) / 14

This simplifies to:

y₁ = 14 / 14

Which gives us:

y₁ = 1

So, our first solution is y₁ = 1. Now, let's move on to the subtraction case:

y₂ = (7 - 7) / 14

This simplifies to:

y₂ = 0 / 14

Which gives us:

y₂ = 0

So, our second solution is y₂ = 0. And there you have it! We've found both solutions to our quadratic equation. Always remember to handle both the addition and subtraction cases to get all possible roots. Now, let's take a moment to summarize our findings and understand what these solutions mean in the context of the original equation.

Summarizing the Solutions

Alright, we've done the work and found our two solutions for the equation 7y² - 7y = 0. Let’s bring it all together and clearly state our answers. We found that:

  • y₁ = 1
  • y₂ = 0

These are the two values of y that make the equation true. In other words, if you substitute either 1 or 0 for y in the original equation, the equation will balance out. You can even try plugging these values back into the original equation to check your work. This is always a good practice to ensure you haven't made any mistakes along the way.

But what do these solutions mean? Graphically, these are the points where the parabola represented by the equation 7y² - 7y intersects the y-axis. They are also sometimes called the roots or zeros of the quadratic equation. Understanding what the solutions represent can give you a deeper insight into the problem. Remember, quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions. In our case, we have two distinct real solutions.

So, to recap, we successfully solved the quadratic equation 7y² - 7y = 0 using the quadratic formula. We identified our coefficients, plugged them into the formula, simplified, and calculated both solutions. Great job, guys! You've conquered another math problem!

Conclusion

So there you have it! We've walked through the process of solving the quadratic equation 7y² - 7y = 0 using the quadratic formula. We started by understanding the formula itself, then carefully identified the coefficients, plugged them in, simplified, and finally calculated our two solutions: y = 1 and y = 0. Remember, the quadratic formula is a powerful tool that can be used to solve any quadratic equation, even those that aren't easily factorable. The key is to take it step by step, paying close attention to detail and avoiding common mistakes like sign errors.

By understanding and practicing with the quadratic formula, you'll be well-equipped to tackle a wide range of quadratic equation problems. And remember, math is a skill that improves with practice, so keep at it! If you ever get stuck, don't hesitate to review the steps, look for examples, or ask for help. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Well done on following along, and I hope this guide has been helpful in demystifying the quadratic formula. Keep up the great work!