Maximizing Quadratic Functions Using Bhaskara's Formula And The Discriminant

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Hey guys! Have you ever wondered how to find the highest point of a parabola? That's where quadratic functions come in! Today, we're going to dive deep into the quadratic function F(x) = -x² + 3x + 28 and figure out its maximum value. We'll be using the famous Bhaskara's formula and the discriminant (delta) to crack this problem. So, buckle up and let's get started!

Understanding Quadratic Functions

Before we jump into solving the problem, let's quickly recap what quadratic functions are all about. A quadratic function is a polynomial function of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic function is:

F(x) = ax² + bx + c

Where 'a', 'b', and 'c' are constants, and 'a' is not equal to 0. The graph of a quadratic function is a parabola, which is a U-shaped curve. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. In our case, F(x) = -x² + 3x + 28, 'a' is -1, which means the parabola opens downwards, and we have a maximum point to find.

The Quest for the Maximum Value

Now, let's get to the heart of the matter: finding the maximum value of F(x) = -x² + 3x + 28. The maximum or minimum value of a quadratic function occurs at the vertex of the parabola. For a parabola that opens downwards (like ours), the vertex represents the highest point, hence the maximum value. There are a couple of ways we can find the vertex, and we'll explore both using Bhaskara's formula and the discriminant.

Bhaskara's Formula: A Quick Refresher

You've probably heard of Bhaskara's formula before. It's a powerful tool for finding the roots (or zeros) of a quadratic equation. The roots are the values of 'x' for which F(x) = 0. Bhaskara's formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

Where:

  • 'a', 'b', and 'c' are the coefficients of the quadratic equation.
  • The expression inside the square root, b² - 4ac, is called the discriminant (Δ or delta).

Decoding the Discriminant (Delta)

The discriminant (Δ) is a crucial part of Bhaskara's formula, and it tells us a lot about the nature of the roots of the quadratic equation. Here's the breakdown:

  • If Δ > 0: The equation has two distinct real roots.
  • If Δ = 0: The equation has one real root (a repeated root).
  • If Δ < 0: The equation has no real roots (two complex roots).

But how does the discriminant help us find the maximum value? Well, it's connected to the vertex! Let's see how.

Finding the Vertex: The Key to Maximum Value

The vertex of the parabola is the point where the function reaches its maximum (or minimum) value. The x-coordinate of the vertex (x_v) can be found using the formula:

x_v = -b / 2a

And the y-coordinate of the vertex (y_v), which is the maximum value of the function, can be found by plugging x_v back into the original function:

y_v = F(x_v)

Alternatively, we can also find the y-coordinate of the vertex (y_v) directly using the discriminant:

y_v = -Δ / 4a

This formula is derived from completing the square and is a handy shortcut. Now, let's apply these formulas to our function.

Cracking the Code: Applying the Formulas to F(x) = -x² + 3x + 28

Alright, let's put our knowledge to the test! Our function is F(x) = -x² + 3x + 28. So, we have:

  • a = -1
  • b = 3
  • c = 28

Step 1: Calculate the Discriminant (Δ)

Let's find the discriminant first:

Δ = b² - 4ac

Δ = (3)² - 4(-1)(28)

Δ = 9 + 112

Δ = 121

Step 2: Find the x-coordinate of the Vertex (x_v)

Now, let's find the x-coordinate of the vertex:

x_v = -b / 2a

x_v = -3 / 2(-1)

x_v = -3 / -2

x_v = 1.5

Step 3: Calculate the y-coordinate of the Vertex (y_v) – The Maximum Value!

We have two ways to find y_v. Let's use both to double-check our answer.

Method 1: Plugging x_v into the Function

y_v = F(x_v) = F(1.5)

y_v = -(1.5)² + 3(1.5) + 28

y_v = -2.25 + 4.5 + 28

y_v = 30.25

Method 2: Using the Discriminant

y_v = -Δ / 4a

y_v = -121 / 4(-1)

y_v = -121 / -4

y_v = 30.25

Both methods give us the same result! So, the maximum value of the function F(x) = -x² + 3x + 28 is 30.25.

The Answer and the Alternatives

Looking at the alternatives provided:

  • A) 29
  • B) 28
  • C) 27
  • D) 30

None of the options exactly match our answer of 30.25. However, option D) 30 is the closest. In a real-world scenario, this might indicate a rounding issue or a slight error in the options. But based on our calculations, 30.25 is the precise maximum value.

Key Takeaways and Real-World Applications

So, what have we learned today? We've mastered the art of finding the maximum value of a quadratic function using Bhaskara's formula and the discriminant. This is a powerful skill that has applications in various fields, such as:

  • Physics: Calculating the maximum height of a projectile.
  • Engineering: Optimizing the design of structures and systems.
  • Economics: Determining the maximum profit or revenue.
  • Computer Graphics: Creating realistic curves and shapes.

Understanding quadratic functions and their properties opens doors to solving a wide range of problems. So, keep practicing, and you'll become a quadratic function whiz in no time!

Final Thoughts

Guys, I hope this step-by-step guide has helped you understand how to find the maximum value of a quadratic function. Remember, the key is to understand the concepts and apply the formulas correctly. Don't be afraid to practice and explore different examples. Math can be fun, especially when you unlock its secrets! Keep learning, keep exploring, and I'll see you in the next math adventure!