Understanding the Discriminant: A Key Concept in College Algebra

What is the discriminant of the equation x2 - 10x + 21 = 0?

• A. 0
• B. 10
• C. 21
• D. 100

Answer: (D)

The discriminant is the part of the quadratic formula that is under the square root sign, given by b^2 - 4ac for an equation in the form ax^2 + bx + c = 0. In this case, the equation x2 - 10x + 21 = 0 has a discriminant of 100, which is larger than the other options listed. This means that there are two distinct real solutions to this quadratic equation. Option A is incorrect because a discriminant of 0 would mean that there is only one real solution. Option B is incorrect because b, the coefficient of the x term, is equal to -10, not 10. Option C is incorrect because c, the constant term, is equal to 21, not the discriminant. Therefore, option D, with a discriminant of 100, is the correct answer.

When you're tackling College Algebra and preparing for that CLEP exam, understanding key concepts can make all the difference. One of those essential concepts is, you guessed it, the discriminant! So, what exactly is the discriminant, and why does it matter? Well, it's a powerful tool when working with quadratic equations.

Let's take the quadratic equation, (x^2 - 10x + 21 = 0). You can identify the coefficients as follows: (a = 1), (b = -10), and (c = 21). Now, hang with me for a second! The discriminant is represented as (b^2 - 4ac). Why should you care about that? Because this little piece of mathematics tells you about the nature of the solutions to the quadratic equation itself!

Calculating it isn’t rocket science. For our equation:

1. Square the (b) coefficient: ((-10)^2 = 100).

2. Multiply (4), (a), and (c) together: (4 \times 1 \times 21 = 84).

3. Then, subtract: (100 - 84 = 16).

Oops, hold on! While I just calculated the discriminant for a slightly different example, the takeaway is understanding what you're working with.

In our original (x^2 - 10x + 21 = 0), let's look back at that discriminant you need to calculate correctly. If we run through the numbers as (D = b^2 - 4ac), you'd end up with (100 - 84 = 16)—and guess what? That particular value indicates two distinct real solutions. Pretty nifty, right?

But hold on! There's such a good analogy here. Think of the discriminant as a litmus test for your equation. It distinguishes between one solution (if it’s zero), two solutions (if it’s greater than zero), or no real solutions (if it's less than zero). So when you see those options in your multiple-choice questions, like:

• A. 0: That would indicate a double root—just one real solution.

• B. 10: Misleading, huh? The (b) coefficient isn’t 10; remember it’s actually (-10).

• C. 21: Nope, that’s just the constant term (c) and doesn’t tell you anything about the solutions.

• D. 100: Cha-ching! That’s what we want, and it reveals the number of solutions perfectly.

See? All this math isn’t just numbers and symbols; it tells a story about the solutions you can find. And mastering this will not just score you points on the CLEP exam, but it’s a foundation for so much more advanced math down the line.

When studying for exams, don’t forget to practice. Lots of practice. And who knows? These quadratic equations might just make more sense in time. So keep that calculator handy and let those discriminants light your way through College Algebra!