Calculating The Discriminant Of A Quadratic Equation

Hey everyone! Today, we're diving into a fundamental concept in algebra: finding the discriminant of a quadratic equation. This might sound a bit intimidating at first, but trust me, it's not that bad. We'll break down the process step by step, and by the end of this, you'll be able to calculate the discriminant with confidence. So, let's get started!

Understanding Quadratic Equations

Before we jump into the discriminant, let's make sure we're all on the same page about quadratic equations themselves. A quadratic equation is simply an equation that can be written in the standard form: ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The 'x' here is our variable, and the equation's solutions (also known as roots or zeros) represent the points where the corresponding parabola (the U-shaped curve that represents the equation graphically) intersects the x-axis. Pretty neat, right?

Think of it like this: the quadratic equation is the blueprint, and the roots are the locations of key features on the graph. The coefficients a, b, and c determine the shape and position of the parabola. The coefficient a dictates whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The b and c coefficients influence the parabola's position and its x-intercepts. Understanding these basics is essential because the discriminant provides critical information about the nature of these roots. Knowing this helps us understand how many real solutions the equation has, or if those solutions are even real numbers at all.

What is the Discriminant?

Alright, so what exactly is the discriminant? In a nutshell, the discriminant is a part of the quadratic formula that gives us crucial information about the roots of a quadratic equation. It's calculated using the formula: D = b² - 4ac. The discriminant is usually represented by the capital letter D. The b, a, and c are coefficients from the standard form of a quadratic equation. The value of the discriminant tells us how many real solutions a quadratic equation has.

• If D > 0: The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If D = 0: The equation has one real root (or two identical real roots). The parabola touches the x-axis at exactly one point (the vertex of the parabola is on the x-axis).
• If D < 0: The equation has no real roots. The parabola does not intersect the x-axis at all (the roots are complex numbers).

Essentially, the discriminant acts as a predictor. It is a tool that allows us to understand the nature of the solutions without actually solving the quadratic equation itself. It is a quick and efficient way to determine the number and type of roots. This is incredibly useful in various mathematical and real-world applications where knowing the nature of the solutions is more important than finding their exact values.

Step-by-Step Calculation: Let's Get to the Problem!

Now, let's get down to brass tacks and solve the problem you gave. Our quadratic equation is –12x² – 18x + 77 = –7x² – 4.

Step 1: Rewrite the Equation in Standard Form

First things first, we need to rewrite our equation into the standard form: ax² + bx + c = 0. To do this, we need to move all the terms to one side of the equation. So let's get all the terms to the left side.

Add 7x² to both sides: -12x² + 7x² – 18x + 77 = –4 This gives us -5x² – 18x + 77 = –4

Next, add 4 to both sides: -5x² – 18x + 77 + 4 = 0

Which simplifies to: -5x² – 18x + 81 = 0

So now our equation is in the correct form, and we can easily identify the coefficients.

Step 2: Identify the Coefficients

Now that the equation is in standard form, we can identify our coefficients:

• a = -5
• b = -18
• c = 81

Remember, these are the values we'll plug into the discriminant formula.

Step 3: Apply the Discriminant Formula

It's time to use the formula: D = b² - 4ac. Let's plug in the values we found in Step 2:

• D = (-18)² - 4 * (-5) * (81)

Step 4: Calculate the Discriminant

Now, let's crunch the numbers:

• D = 324 - (-1620)
• D = 324 + 1620
• D = 1944

So, the discriminant (D) is equal to 1944.

Interpreting the Result

Great job! We've successfully calculated the discriminant. But what does a discriminant of 1944 tell us? Since D > 0 (1944 is greater than zero), we know that the quadratic equation has two distinct real roots. This means that if we were to graph the equation, the parabola would cross the x-axis at two different points. This is an important piece of information since it tells us the type of solutions we can expect without actually having to solve the equation itself. Isn't that neat?

This simple calculation gives us valuable insight into the behavior of the quadratic equation. Whether you're working on a math problem, or using these concepts in real-world applications (like physics or engineering), the discriminant is a powerful tool to have in your arsenal. The discriminant acts like a quick diagnostic tool, offering insights into the nature of the solutions. This is useful for problem-solving.

Conclusion: You Got This!

And there you have it! We've successfully calculated the discriminant of a quadratic equation. Remember, the discriminant is your friend! It helps you understand the nature of the roots without having to fully solve the equation. The discriminant tells you if you have two real roots, one real root (or two identical), or no real roots (complex roots). Understanding these basics is critical for a solid foundation in algebra. Keep practicing, and you'll become a pro in no time.

I hope this step-by-step guide has been helpful. If you have any more questions, feel free to ask. Keep up the great work, and happy calculating!