Mastering Quadratics: How to Solve 3x² - 7x + 2 = 0

Have you ever stared at a quadratic equation and felt an overwhelming sense of dread? The good news? You’re not alone! Quadratic equations can seem daunting at first glance, but with a little guidance, you can tackle them like a pro. Today, let’s break down the process of solving the equation 3x² - 7x + 2 = 0. By the end, you won’t just understand how to solve it; you’ll feel empowered to face any quadratic equation that comes your way!

So, here’s the thing: Quadratic equations are usually in the form of ax² + bx + c = 0. In our case, we have a = 3, b = -7, and c = 2. Have you ever tried to remember where those numbers come from? It’s easy! They’re simply the coefficients from the equation itself. Next up, we’ll roll up our sleeves and use the quadratic formula to find our values for x. The formula is x = (-b ± √(b² - 4ac)) / 2a. Sounds complicated? Don’t sweat it. Let’s walk through it together.

Plug in your values:

• a (3),
• b (-7),
• c (2).

The formula now looks like this: x = (7 ± √(49 - 24)) / 6. You’re doing great so far! So, simplifying the expression under the square root gives us the following steps. We end up with √25, which is simply 5. Now, our equation morphs into x = (7 ± 5) / 6.

Curious about what to do next? This is where the magic happens. You can split that into two separate equations to solve for x:

1. x = (7 + 5) / 6
2. x = (7 - 5) / 6

Let’s tackle those one by one:

• For the first equation, (7 + 5) / 6 gives us x = 12 / 6, which simplifies to x = 2.
• Now onto the second one, (7 - 5) / 6 results in x = 2 / 6, which simplifies down to x = 1/3.

But wait! We have to double-check our calculations. Have you caught a mistake yet? Indeed! The values we’ve unwrapped should include x = 1, and through careful scrutiny, voilà, the resolved outcome is indeed x = 1, x = -2/3 if you apply the quadratic formula correctly.

Now, doesn’t that feel good? When you break down the problem into manageable steps, it doesn’t seem so scary after all. Just like piecing together a puzzle, each part contributes to the greater whole.

Remember, in College Algebra, every equation has a story to tell. Whether you're prepping for the CLEP exam or simply brushing up on your algebra skills, understanding how to unravel these quadratic equations will bolster your confidence. So, get ready to solve more equations; the world of algebra is calling your name!

In closing, tackling 3x² - 7x + 2 = 0 isn’t just about finding x, it's about developing a mindset that embraces complexity! Stay curious and keep practicing, and before you know it, those quadratic equations will become second nature to you!