Calculating The Discriminant: B² - 4ac For X² + 5x + 4 = 0

Hey guys! Ever wondered how to figure out the nature of the roots of a quadratic equation without actually solving for them? Well, the discriminant is your answer! It's a super handy tool in algebra, and today, we're going to dive deep into calculating it. We'll take the quadratic equation $x^2 + 5x + 4 = 0$ as our example and break down each step. So, grab your calculators (or just your brain!), and let's get started!

Understanding the Discriminant

At its core, the discriminant is a part of the quadratic formula that reveals a ton about the roots (or solutions) of a quadratic equation. You might remember the quadratic formula itself:

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

See that little gem under the square root, $b^2 - 4ac$? That's our star – the discriminant! It's so important because its value tells us whether the quadratic equation has two distinct real roots, one repeated real root, or two complex roots. Think of it like a sneak peek into the solution landscape of the equation. Now, let's really break down why this works. The discriminant, mathematically represented as $\Delta = b^2 - 4ac$, acts as a critical indicator of the nature and number of roots in a quadratic equation. When $\Delta > 0$, it signals that the equation has two distinct real roots. Real roots are solutions that are real numbers, and having two distinct ones means you'll find two different values of x that satisfy the equation. This happens because the square root of a positive number yields two real values (one positive and one negative), leading to two different solutions when applied in the quadratic formula. In contrast, when $\Delta = 0$, the equation has exactly one real root, which we refer to as a repeated or double root. This scenario occurs because the square root of zero is zero, thus eliminating the $\pm$ part of the quadratic formula. As a result, both parts of the formula collapse into a single solution. Lastly, when $\Delta < 0$, the equation has two complex roots. Complex roots involve the imaginary unit i, which is defined as the square root of -1. Since you can't take the square root of a negative number in the real number system, the roots enter the complex number realm. These complex roots always come in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root. Understanding these distinctions is crucial for solving quadratic equations effectively, as it helps in predicting the type and number of solutions to expect.

Identifying a, b, and c

Okay, before we can actually calculate the discriminant, we need to identify the coefficients a, b, and c from our equation: $x^2 + 5x + 4 = 0$. Remember the standard form of a quadratic equation is:

$ax^2 + bx + c = 0$

Now, let's match the coefficients:

• The coefficient of $x^2$ is a. In our case, $a = 1$ (since $x^2$ is the same as $1x^2$).
• The coefficient of $x$ is b. Here, $b = 5$.
• The constant term is c. So, $c = 4$.

See? It's like a little puzzle! Now that we've got our a, b, and c values, we're ready to plug them into the discriminant formula. Once you've identified the coefficients a, b, and c, you're essentially setting the stage for the rest of the problem. This step is so crucial because any mistake here will throw off your entire calculation. Think of it as laying the foundation of a building—if it's not solid, everything else is at risk. To make sure we're rock solid, let's go through a few examples beyond our initial equation. For the quadratic equation $2x^2 - 3x + 1 = 0$, we have a as 2, b as -3, and c as 1. Notice how paying attention to the signs is essential, especially when b is negative. Similarly, in the equation $x^2 + 4x - 7 = 0$, a is 1, b is 4, and c is -7. Another example could be $-x^2 + 6x - 9 = 0$, where a is -1, b is 6, and c is -9. By practicing this identification with various equations, you'll become super confident in picking out these coefficients. This not only speeds up your problem-solving process but also minimizes the chances of making errors, making it a win-win situation! Remember, the key is to meticulously compare the given equation with the standard form and ensure that every coefficient is correctly matched with its corresponding term.

Plugging into the Formula

Alright, now for the fun part: plugging our values into the discriminant formula! We know:

• $b = 5$
• $a = 1$
• $c = 4$

And our formula is:

$\Delta = b^2 - 4ac$

So, let's substitute:

$\Delta = 5^2 - 4 * 1 * 4$

See how we just replaced the letters with the numbers? Now it's just a matter of doing the math. This step, where you plug the values into the formula, is like the bridge between recognizing the components and getting to the solution. It’s crucial to be methodical here to avoid any errors in substitution. Think of it as assembling a puzzle—you’ve identified the pieces, and now you’re carefully fitting them together. When substituting, it’s a good practice to rewrite the formula each time, followed by the values. This not only reinforces the formula in your mind but also reduces the chances of making a mistake. For example, writing “$\Delta = b^2 - 4ac = (5)^2 - 4(1)(4)$” makes it clear what values you’re using and where they’re going. It's also helpful to use parentheses around the values, especially when dealing with negative numbers, to ensure you’re handling the signs correctly. So, if you were substituting b with -5, writing “(-5)^2” clarifies that you’re squaring -5, which results in 25, rather than squaring 5 and then negating it, which would incorrectly give you -25. This attention to detail can make a significant difference in your calculations. Additionally, keeping track of the order of operations (PEMDAS/BODMAS) is vital to ensure the correct calculation of the discriminant. By being careful and systematic in this substitution step, you’re setting yourself up for success in the rest of the problem.

Calculating the Result

Time to crunch some numbers! Let's continue from where we left off:

$\Delta = 5^2 - 4 * 1 * 4$

First, we handle the exponent:

$\Delta = 25 - 4 * 1 * 4$

Next, we perform the multiplication from left to right:

$\Delta = 25 - 16$

Finally, the subtraction:

$\Delta = 9$

Woohoo! We've got our discriminant. So, for the equation $x^2 + 5x + 4 = 0$, the value of $b^2 - 4ac$ is 9. Now, calculating the discriminant is like decoding a secret message from the quadratic equation. Each step requires precision, like a detective piecing together clues. Starting from the substitution, you're setting the stage for the final reveal. When you hit the exponentiation, remember that $5^2$ means 5 times 5, not 5 times 2. This might seem basic, but it's a common slip-up that can throw off your whole calculation. Next, you dive into the multiplication. The order matters here—4 times 1 times 4 is different from 4 times (1 times 4) only in how you think about it, but ensuring you get each multiplication correct sets you up for the final subtraction. Then comes the grand finale: the subtraction itself. This is where all the previous steps culminate into a single number—the discriminant. If you've been careful with each step, this number will tell you a lot about the roots of the equation. If we made a mistake along the way, the final number will be wrong and the interpretation will be wrong. So, double-checking your calculations is like making sure all your clues lead to the right suspect. And once you arrive at the final value, you can confidently say,