Solving 0.2x^2 - 0.8x - 0.2 = 0 With Quadratic Formula

Hey guys! Ever get stuck with a quadratic equation that just won't factor? Don't sweat it! The quadratic formula is your superhero in disguise, always ready to save the day. In this article, we're going to break down how to use the quadratic formula to solve the equation 0.2x^2 - 0.8x - 0.2 = 0. So, let's dive in and make math a little less intimidating, shall we?

Understanding the Quadratic Formula

Before we jump into solving our specific equation, let's quickly recap what the quadratic formula actually is. The quadratic formula is a powerful tool used to find 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 of ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The quadratic formula is:

x = [-b ± √(b^2 - 4ac)] / (2a)

This formula might look a bit intimidating at first glance, but trust me, it's not as scary as it seems. It's just a matter of plugging in the right numbers and doing some arithmetic. The key here is to correctly identify the values of a, b, and c from your equation. Once you've got those, it's smooth sailing! The ± symbol simply means there are usually two solutions, one where you add the square root part and one where you subtract it. Think of it as a clever way to write two separate equations in one.

Now, why is the quadratic formula so important? Well, it's a universal key that unlocks the solutions to any quadratic equation, even those that are difficult or impossible to factor. Factoring is a great method when it works, but it's not always practical. The quadratic formula is like your reliable backup plan, always there when you need it. Plus, understanding and using it gives you a deeper insight into the nature of quadratic equations and their solutions. So, buckle up, and let's get those quadratic equations solved!

Identifying a, b, and c in Our Equation

Okay, let's get down to business with our equation: 0.2x^2 - 0.8x - 0.2 = 0. The first step in using the quadratic formula is to correctly identify the values of a, b, and c. Remember the standard form of a quadratic equation: ax^2 + bx + c = 0. Matching this form to our equation, we can easily pick out the coefficients.

Here's how it breaks down:

• The coefficient of the x^2 term is 'a'. In our equation, a = 0.2.
• The coefficient of the x term is 'b'. Here, b = -0.8. Don't forget the negative sign! This is a common mistake, so always double-check your signs.
• The constant term is 'c'. In this case, c = -0.2. Again, make sure you include the negative sign.

So, to recap, we have:

• a = 0.2
• b = -0.8
• c = -0.2

Now that we have these values, we're ready to plug them into the quadratic formula. This is a crucial step, and getting these values right is half the battle. Once you have a, b, and c correctly identified, the rest is just careful calculation. Trust me; you've got this! Next up, we'll substitute these values into the formula and start crunching those numbers. Stay tuned!

Plugging the Values into the Quadratic Formula

Alright, guys, we've identified a, b, and c – now comes the fun part: plugging those values into the quadratic formula! Remember the formula? Let's write it down again to keep it fresh in our minds:

x = [-b ± √(b^2 - 4ac)] / (2a)

We found that a = 0.2, b = -0.8, and c = -0.2. Now, we carefully substitute these values into the formula. This is where attention to detail is key. A small mistake here can throw off the entire solution, so let's take our time and do it right.

Replacing the variables with their values, we get:

x = [-(-0.8) ± √((-0.8)^2 - 4 * 0.2 * (-0.2))] / (2 * 0.2)

Notice how I've put the negative values in parentheses? This is super important, especially when dealing with squares and subtractions. It helps avoid sign errors, which are very common. Okay, let's simplify this step by step.

First, -(-0.8) becomes 0.8. So, we have:

x = [0.8 ± √((-0.8)^2 - 4 * 0.2 * (-0.2))] / (2 * 0.2)

Next, let's deal with the expression inside the square root. We'll calculate (-0.8)^2, 4 * 0.2 * (-0.2), and then subtract. This is all about following the order of operations (PEMDAS/BODMAS). So, let's keep going and break it down further!

Simplifying the Equation

Okay, let's continue simplifying our equation. We're at:

x = [0.8 ± √((-0.8)^2 - 4 * 0.2 * (-0.2))] / (2 * 0.2)

First, let's tackle the square inside the square root: (-0.8)^2. Remember, a negative number squared becomes positive. So, (-0.8)^2 = 0.64.

Now our equation looks like this:

x = [0.8 ± √(0.64 - 4 * 0.2 * (-0.2))] / (2 * 0.2)

Next, let's handle the multiplication part: 4 * 0.2 * (-0.2). This gives us -0.16. So, we now have:

x = [0.8 ± √(0.64 - (-0.16))] / (2 * 0.2)

Subtracting a negative is the same as adding, so 0.64 - (-0.16) becomes 0.64 + 0.16, which equals 0.8.

Our equation is now:

x = [0.8 ± √(0.8)] / (2 * 0.2)

Finally, let's simplify the denominator: 2 * 0.2 = 0.4.

So, we have:

x = [0.8 ± √(0.8)] / 0.4

We've made some great progress! The equation is looking much simpler now. Next, we'll calculate the square root of 0.8 and then find our two possible solutions for x.

Calculating the Solutions

Alright, guys, we're in the home stretch! Our equation is now simplified to:

x = [0.8 ± √(0.8)] / 0.4

Let's start by finding the square root of 0.8. You'll probably want to use a calculator for this. √0.8 is approximately 0.894 (rounded to three decimal places).

Now we have:

x = [0.8 ± 0.894] / 0.4

Remember that ± sign? It means we actually have two separate equations to solve:

1. x = (0.8 + 0.894) / 0.4
2. x = (0.8 - 0.894) / 0.4

Let's solve the first one. 0.8 + 0.894 = 1.694. So, x = 1.694 / 0.4. Dividing 1.694 by 0.4 gives us approximately 4.235.

So, one solution is x ≈ 4.235.

Now, let's solve the second equation. 0.8 - 0.894 = -0.094. So, x = -0.094 / 0.4. Dividing -0.094 by 0.4 gives us -0.235.

So, our second solution is x ≈ -0.235.

Therefore, the solutions to the quadratic equation 0.2x^2 - 0.8x - 0.2 = 0 are approximately x = 4.235 and x = -0.235.

Conclusion

And there you have it! We've successfully used the quadratic formula to solve the equation 0.2x^2 - 0.8x - 0.2 = 0. It might seem like a lot of steps, but each one is manageable when you break it down. Remember, the key is to carefully identify a, b, and c, plug them into the formula accurately, and then take your time simplifying.

The quadratic formula is a fantastic tool for solving quadratic equations, especially those that don't factor easily. So next time you're faced with a quadratic equation, don't panic! Just remember the quadratic formula, and you'll be solving like a pro in no time. Keep practicing, and you'll master this skill in no time. You got this, guys! And if you ever get stuck, just revisit this guide or reach out for help. Happy solving!