Find The Zeros Of Quadratic Function F(b) = B^2 + 10b + 16

Hey everyone! Today, we're going to dive into the world of quadratic functions and learn how to find their zeros. Specifically, we'll be tackling the function f(b) = b^2 + 10b + 16. Finding the zeros of a function is a fundamental concept in algebra, and it's super useful in various real-world applications, from physics to engineering to economics. So, let's get started and break this down step by step!

What are Zeros of a Function?

First off, let's clarify what we mean by the "zeros" of a function. Simply put, the zeros of a function are the values of the input variable (in our case, b) that make the function equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. Think of it like finding the spots where the function 'flatlines' at zero. These points are also known as roots or x-intercepts.

For the quadratic function f(b) = b^2 + 10b + 16, we are looking for the values of b that satisfy the equation b^2 + 10b + 16 = 0. There are several methods to find these zeros, including factoring, completing the square, and using the quadratic formula. We'll focus on factoring because it's often the quickest and most straightforward method when applicable. Factoring involves breaking down the quadratic expression into a product of two binomials. When we have the expression in factored form, we can easily identify the values of b that make each binomial equal to zero, which are the zeros of the function.

Why is this important, you ask? Well, zeros can represent critical points in a system. For instance, in a projectile motion problem, the zeros might represent the times when the projectile hits the ground. In business, they might represent break-even points where costs equal revenue. Understanding how to find these zeros allows us to analyze and solve various real-world problems, making this a crucial skill in mathematics and beyond. So, let's get our hands dirty and apply this to our function!

Factoring the Quadratic Expression

Factoring is a powerful technique to find the zeros of a quadratic function. To factor the quadratic expression b^2 + 10b + 16, we need to find two numbers that multiply to 16 (the constant term) and add up to 10 (the coefficient of the b term). Let's think about the factors of 16: 1 and 16, 2 and 8, 4 and 4. Which pair adds up to 10? You guessed it – 2 and 8!

So, we can rewrite the quadratic expression as a product of two binomials: (b + 2)(b + 8). This means that b^2 + 10b + 16 = (b + 2)(b + 8). The expression is now in a factored form, which is super handy for finding the zeros. Factoring transforms a seemingly complex quadratic into a manageable form where we can directly identify the roots. It's like having a secret key that unlocks the solution!

The beauty of factoring lies in its simplicity and directness. Once you identify the correct factors, you're just a small step away from finding the zeros. The factored form gives us a clear picture of what values of b will make the expression equal to zero. It's a bit like solving a puzzle where each piece (the factors) fits perfectly to reveal the complete picture (the zeros). So, now that we have factored the expression, let's proceed to find those zeros and see where our function intersects the x-axis!

Finding the Zeros

Now that we have factored the quadratic expression f(b) = b^2 + 10b + 16 into (b + 2)(b + 8), we are ready to find the zeros. Remember, the zeros are the values of b that make the function equal to zero. So, we need to solve the equation (b + 2)(b + 8) = 0. This is where the zero-product property comes into play. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if A * B* = 0, then either A = 0 or B = 0 (or both).

Applying this property to our factored expression, we set each factor equal to zero and solve for b:

1. b + 2 = 0

   Subtracting 2 from both sides, we get b = -2.

2. b + 8 = 0

   Subtracting 8 from both sides, we get b = -8.

So, we have found two values of b that make the function f(b) equal to zero: b = -2 and b = -8. These are the zeros of the function. It’s like finding the hidden switches that turn the function off, or in mathematical terms, bring it down to zero. These zeros tell us where the graph of the quadratic function crosses the x-axis, giving us crucial points for understanding the function's behavior.

Graphical Interpretation

Understanding the graphical interpretation of these zeros can provide even deeper insights. The zeros, b = -2 and b = -8, are the x-intercepts of the parabola represented by the quadratic function f(b) = b^2 + 10b + 16. This means the parabola crosses the x-axis at the points (-2, 0) and (-8, 0). Visualizing this can be incredibly helpful. Imagine a U-shaped curve (a parabola) that dips below the x-axis and then crosses it at -8, continues upwards, turns around, and crosses the x-axis again at -2.

The x-intercepts are not just random points; they are critical landmarks on the graph. They help us understand the shape and position of the parabola. The axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves, will pass through the midpoint of the zeros. In this case, the axis of symmetry is b = -5. The vertex, which is the minimum or maximum point of the parabola, lies on this axis of symmetry. This gives us a clear picture of how the function behaves and where its turning point is located.

Furthermore, the zeros divide the x-axis into intervals where the function is either positive or negative. Between -8 and -2, the function is negative (the parabola is below the x-axis), and outside this interval (less than -8 and greater than -2), the function is positive (the parabola is above the x-axis). This understanding is crucial in solving inequalities and analyzing the function’s overall behavior. So, the zeros are not just solutions; they are keys that unlock a wealth of information about the function’s graphical representation and its properties.

Conclusion

In summary, we successfully found the zeros of the quadratic function f(b) = b^2 + 10b + 16. By factoring the quadratic expression into (b + 2)(b + 8) and applying the zero-product property, we determined that the zeros are b = -2 and b = -8. These zeros represent the points where the function's graph intersects the x-axis, giving us valuable insights into the function's behavior and graphical representation.

Finding the zeros of a function is a fundamental skill in algebra with wide-ranging applications. Whether you're solving projectile motion problems, analyzing business scenarios, or simply trying to understand the behavior of a mathematical function, knowing how to find the zeros is essential. The methods we've discussed, particularly factoring, are powerful tools in your mathematical toolkit.

So, next time you encounter a quadratic function, remember the steps: factor the expression, apply the zero-product property, and solve for the zeros. You'll be well-equipped to tackle a variety of problems and deepen your understanding of mathematical functions. Keep practicing, and you'll become a pro at finding those zeros! Great job, everyone!