Solving $3x^2 - 10x + 5 = 0$ With Quadratic Formula

Hey guys! Today, we're diving into the world of quadratic equations and tackling the equation $3x^2 - 10x + 5 = 0$. Now, when we're faced with a quadratic equation like this, one of the most reliable tools in our arsenal is the quadratic formula. It might seem a bit intimidating at first, but trust me, once you get the hang of it, it's super useful! Let's break it down step by step, so you can see exactly how it works and why it's so effective. We'll start by understanding what the quadratic formula actually is and then apply it to our specific equation. Ready? Let's jump in!

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions (also called roots or zeros) of any quadratic equation. A quadratic equation is an equation that can be written in the standard form:

$ax^2 + bx + c = 0$

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

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Okay, let's dissect this formula piece by piece. The ± symbol means we'll actually get two solutions, one using the plus sign and one using the minus sign. The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us a lot about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have exactly one real solution (a repeated root). And if it's negative, we have two complex solutions. Knowing this can give you a heads-up on what kind of answers to expect before you even plug in the numbers! This formula might look a little scary at first glance, but it's really just a matter of plugging in the correct values and doing the arithmetic. The key is to correctly identify a, b, and c from your quadratic equation. Once you've got those down, the rest is just following the steps. Think of it as a recipe – follow the instructions, and you'll get the right result every time. Let's move on to applying this formula to our equation, so you can see exactly how it works in practice.

Applying the Quadratic Formula to $3x^2 - 10x + 5 = 0$

Now, let's apply this quadratic formula to our specific equation: $3x^2 - 10x + 5 = 0$. The first thing we need to do is identify the coefficients a, b, and c. Comparing our equation to the standard form $ax^2 + bx + c = 0$, we can see that:

• a = 3
• b = -10
• c = 5

Make sure you pay close attention to the signs! A negative sign in front of a coefficient is super important and can totally change your answer if you miss it. Now that we've identified a, b, and c, we can plug these values into the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substituting our values, we get:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(5)}}{2(3)}$

Okay, time to simplify! Let's start with the easier parts. The $-(-10)$ becomes $+10$, and $2(3)$ becomes $6$. So we have:

$x = \frac{10 \pm \sqrt{(-10)^2 - 4(3)(5)}}{6}$

Next, we need to tackle the expression under the square root. $(-10)^2$ is $100$, and $4(3)(5)$ is $60$. So our equation becomes:

$x = \frac{10 \pm \sqrt{100 - 60}}{6}$

Subtracting inside the square root, we get:

$x = \frac{10 \pm \sqrt{40}}{6}$

Alright, we're getting there! Now, let's simplify the square root. We know that $40$ can be written as $4 imes 10$, and $4$ is a perfect square. So, $\sqrt{40}$ can be simplified as $\sqrt{4 imes 10} = \sqrt{4} imes \sqrt{10} = 2\sqrt{10}$. Substituting this back into our equation, we have:

$x = \frac{10 \pm 2\sqrt{10}}{6}$

Finally, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $2$. Dividing each term by $2$, we get:

$x = \frac{5 \pm \sqrt{10}}{3}$

And that's our final answer! We've successfully used the quadratic formula to find the solutions to the equation $3x^2 - 10x + 5 = 0$. See? It wasn't so bad after all. The key is to take it step by step, keep track of your signs, and simplify as much as you can. We're almost there, but before we wrap up, let's quickly review our steps and talk about how to check your answers to make sure you got it right. Trust me, taking that extra step to check your work can save you a lot of headaches!

Checking the Solutions and Final Answer

So, after all that work, we've arrived at our solutions: $x = \frac{5 \pm \sqrt{10}}{3}$. But how can we be sure these are the correct solutions? Well, the best way to check is to plug these values back into the original equation and see if they make it true. It might sound like a bit of a hassle, but it's a surefire way to catch any mistakes you might have made along the way. Remember, it's always better to be safe than sorry! Let's quickly recap the steps we took to get here, just to make sure everything is crystal clear.

First, we identified the coefficients a, b, and c from the equation $3x^2 - 10x + 5 = 0$. We found that a = 3, b = -10, and c = 5. Then, we plugged these values into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. After substituting and simplifying, we got $x = \frac{10 \pm \sqrt{40}}{6}$. We further simplified the square root and the fraction to arrive at our final answer: $x = \frac{5 \pm \sqrt{10}}{3}$. Now, let's talk about what this answer actually means. We have two solutions here, one with the plus sign and one with the minus sign. These are the two values of x that make the equation $3x^2 - 10x + 5 = 0$ true. Graphically, these solutions represent the points where the parabola described by the equation intersects the x-axis. Cool, right? Knowing the solutions allows us to understand the behavior of the quadratic equation and its corresponding graph. It's not just about getting the numbers; it's about understanding what those numbers represent. And that, my friends, is the real power of math! You can also double-check your solutions using online calculators or graphing tools to confirm your results. This can be especially helpful if you're still getting comfortable with the process or if you just want an extra layer of certainty. Always remember, practice makes perfect. The more you work with the quadratic formula, the more comfortable and confident you'll become with it. So, don't be afraid to tackle more problems and explore different types of quadratic equations. You've got this!

Conclusion

Alright guys, we've reached the end of our journey through the quadratic formula! We've seen how to use it to solve the equation $3x^2 - 10x + 5 = 0$, and we've talked about why this formula is such a valuable tool in mathematics. We started by understanding the quadratic formula itself, breaking down each part and seeing how it all fits together. Then, we applied it to our specific equation, carefully plugging in the values and simplifying step by step. And finally, we discussed how to check our solutions to make sure we got it right. The quadratic formula might seem a little intimidating at first, but once you get the hang of it, it's like having a superpower for solving quadratic equations. It's a reliable and versatile tool that can help you tackle a wide range of problems. So, the next time you're faced with a quadratic equation, don't panic! Just remember the formula, take it step by step, and you'll be able to find the solutions with confidence. Keep practicing, keep exploring, and most importantly, keep having fun with math! Thanks for joining me on this adventure, and I'll catch you in the next one. Keep shining, Plastik Magazine readers!