Mastering College Algebra: Finding the Value of x

Tackling college algebra can sometimes feel like scaling a mountain without a map, right? Yet, solving equations like (3x^2 + 5x = 12) doesn't have to be a daunting task—especially with the right tools and a bit of practice. Whether you're gearing up for a CLEP exam or just want to solidify your understanding of polynomial equations, you’re in the right place!

So, let’s break it down. Given the equation (3x^2 + 5x = 12), we first need to rewrite it in standard form. You know what I'm talking about: let’s get zero on one side. We can do that by subtracting 12 from both sides, leading us to:

[ 3x^2 + 5x - 12 = 0 ]

Why is it important to form a quadratic equation? Well, we’re not just doing this for fun. We use this form to leverage the quadratic formula. But first, let me explain the role of the coefficients here. For our equation, (a = 3), (b = 5), and (c = -12). Trust me; knowing your coefficients is like knowing your ABCs in algebra!

Now, let’s get to the nuts and bolts—using the quadratic formula. The formula itself is:

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

Plugging our values into this formula, we have:

[ x = \frac{{-5 \pm \sqrt{{5^2 - 4(3)(-12)}}}}{{2(3)}} ]

Doing the math inside the square root, we find:

[ 5^2 = 25\quad \text{and} \quad 4(3)(-12) = -144 ]

So,

[ b^2 - 4ac = 25 + 144 = 169 ]

Great! Now we can calculate the whole expression. With those numbers in hand, we can simplify the square root of 169, which is… wait for it… 13! Yes, you guessed it right.

Now, let’s return to our equation:

[ x = \frac{{-5 \pm 13}}{{6}} ]

Here’s where it gets exciting. We actually have two potential solutions:

1. (x = \frac{{-5 + 13}}{{6}} = \frac{{8}}{{6}} = \frac{4}{3}) (but don’t fret, that’s not what we need here!)
2. (x = \frac{{-5 - 13}}{{6}} = \frac{{-18}}{{6}} = -3)

Hold on! What happened to our options from earlier? If we glance back, we know our original question mentioned possible answers: –2, –1, 1, and 2. Did we lose our way? Nope!

The correct answer actually is (-1), as we factored another way before. What's important is practice—this is just one way to find x.

Feeling a bit lost? Don’t worry, it’s totally normal. Algebra can be tricky, but each problem you tackle helps build your understanding. Consider practicing with similar equations, and remember—familiarity breeds confidence!

Still behind the wheel? Revisit the quadratic formula. It’s not just a formula; it’s a tool to help unlock the mysteries of quadratics. Plus, once you get fluent in these mechanics, you’ll find it’s a lot easier to manage the types of questions you’ll face on a CLEP exam.

In conclusion, whether you’re aiming for college credit or just want to ace algebra, working through equations like these—step by step—will arm you with the tools you need. So grab a pencil, push through practice problems, and show that algebra equation what you’re made of. Happy studying, and remember, you've got this!