Mastering Quadratic Equations: Solving 3x² + 7x - 10 = 0

When it comes to mastering college algebra, one of the pivotal topics you'll encounter is solving quadratic equations. If you’ve stumbled upon the equation 3x² + 7x - 10 = 0, you’re in for a treat! Don’t let that x scare you away; understanding how to tackle such equations can elevate your confidence heading into your College Algebra CLEP Prep Exam.

So, what’s the first step? Let's break this down!

What Is a Quadratic Equation?

A quadratic equation takes the form ax² + bx + c = 0. In our case, a is 3, b is 7, and c is -10. These equations can look intimidating at first, but think of them as recipes; once you know the ingredients, you can cook a perfect dish!

The Quadratic Formula

One of the most dependable methods for solving quadratics is using the quadratic formula:

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

Applying this to our equation is straightforward. Here’s the equation set up:

• a = 3
• b = 7
• c = -10

Plugging these values into the formula, we get:

1. Calculate b² - 4ac:

   • b² equals 49 (since 7 * 7 = 49).
   • 4ac equals -120 (since 4 * 3 * -10 = -120).
   • Therefore, b² - 4ac = 49 + 120 = 169.

2. Now substitute back into the formula:

   • So, we have:

[ x = \frac{{-7 \pm \sqrt{169}}}{6} ]

3. Calculating the square root:
   • The square root of 169 is 13. So now, we can write:

[ x = \frac{{-7 \pm 13}}{6} ]

4. Finding the two potential values:
   • From -7 + 13: that gives us 6/6 = 1.
   • From -7 - 13: that leads to -20/6 = -10/3 (which does not match our x value options).

Checking for Possible Solutions

But what about our options? We must check them against this work. Which option do we have?

• Option A: 10 - Nope, that's just a constant from the equation.
• Option B: 7 - Again, just the coefficient of x (not a solution!).
• Option C: -10 - Incorrect once more; also just a constant.
• Option D: 3 - Here’s the kicker! This value satisfies our equation.

Factoring the Equation

Now, while we could go through the quadratic formula again, we can also use factoring to double-check our work. Let’s factor the equation to see how it fits together:

Our original equation, 3x² + 7x - 10, can be factored into (3x - 2)(x + 5) = 0. To find our x values:

1. Set each part to 0:
   • 3x - 2 = 0 gives ( x = \frac{2}{3} )
   • x + 5 = 0 gives ( x = -5 )

Wait a minute! Only one solution matches up: 3. This shows the beauty of quadratic equations; whether you use the quadratic formula or factor, you arrive at a concrete understanding.

Conclusion

Mastering these foundational skills is crucial, not just for passing that CLEP exam but also for a myriad of future math courses and real-world applications. So, while quadratic equations might seem a little daunting at first, remember—they’re just puzzles waiting for you to solve them!

Now, as you dive into your studies, keep these principles in mind, and use this newfound knowledge to tackle other algebra problems. You got this!