Discover the Roots of f(x) = x² – 2x – 3: What's the Missing Number?

The roots of the function f(x) = x^2 – 2x – 3 are essential in understanding its behavior and properties. When we solve for the roots, we find that x = –1 and x = [missing number]. These values play a crucial role in determining the critical points, symmetry, and overall shape of the graph. By exploring the missing number, we can unravel more insights into the function and its relationship with the real number line. Let us delve deeper into this intriguing mathematical puzzle.

The Roots of the Function F(x) = x^2 - 2x - 3 are Shown

When analyzing a quadratic function, it is crucial to understand its roots, which represent the values of x that make the function equal to zero. The roots provide valuable information about the behavior and properties of the function. In this article, we will explore the roots of the function F(x) = x^2 - 2x - 3 and determine the missing number when x = -1.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, represented in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, and it can have zero, one, or two real roots depending on the discriminant, which is given by b^2 - 4ac.

Finding the Roots of F(x) = x^2 - 2x - 3

To find the roots of the quadratic function F(x) = x^2 - 2x - 3, we need to set the function equal to zero and solve for x. So, we have the equation x^2 - 2x - 3 = 0.

Applying the Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It states that the roots of a quadratic equation ax^2 + bx + c = 0 are given by x = (-b ± √(b^2 - 4ac)) / (2a).

Substituting the Values

By comparing the equation x^2 - 2x - 3 = 0 with the general form ax^2 + bx + c = 0, we can determine that a = 1, b = -2, and c = -3. Substituting these values into the quadratic formula, we get x = (-(-2) ± √((-2)^2 - 4(1)(-3))) / (2(1)).

Simplifying the Equation

Let's simplify the equation further. We have x = (2 ± √(4 + 12)) / 2, which simplifies to x = (2 ± √16) / 2.

Calculating the Roots

Now, let's calculate the roots of the equation. For x = (2 + √16) / 2, we have x = (2 + 4) / 2, which equals 3. And for x = (2 - √16) / 2, we have x = (2 - 4) / 2, which equals -1.

Identifying the Missing Number

From our calculations, we found that one of the roots of the function F(x) = x^2 - 2x - 3 is x = -1. However, we still need to determine the missing number when x = ?. To find this missing number, we need to refer back to the original equation.

Plugging in x = -1

If we substitute x = -1 into the function F(x) = x^2 - 2x - 3, we get F(-1) = (-1)^2 - 2(-1) - 3, which simplifies to F(-1) = 1 + 2 - 3 = 0.

The Missing Number is 0

Therefore, the missing number when x = -1 in the function F(x) = x^2 - 2x - 3 is 0. This means that when x equals -1, the function evaluates to zero.

In Conclusion

The roots of a quadratic function provide us with essential insights into its behavior and properties. By solving the equation x^2 - 2x - 3 = 0, we found that the roots are x = 3 and x = -1. Plugging x = -1 into the function, we determined that the missing number is 0. Understanding the roots of a function allows us to analyze its graph, determine its vertex, and solve various real-world problems effectively.

Introduction: Unveiling the roots of the function f(x) = x² - 2x - 3

The function f(x) = x² - 2x - 3 is a quadratic function, represented by a parabolic curve in a coordinate system. The roots of a function are the values of x at which the function equals zero. These roots play a crucial role in understanding the behavior and properties of the function. In this paragraph, we will explore the roots of the given function and determine the missing number.

Definition of roots: Roots represent the values of x at which a function equals zero

In order to understand the concept of roots, it is important to comprehend their significance in relation to the function itself. When a function equals zero, it means that the output or y-value is equal to zero. In terms of the given function f(x) = x² - 2x - 3, the roots represent the specific values of x for which f(x) becomes zero. By finding these roots, we can gain insight into the behavior and characteristics of the function.

Unveiling the first root: The first value of x, -1, at which f(x) equals zero

One of the roots of the function f(x) = x² - 2x - 3 has already been provided as x = -1. This means that when x is equal to -1, the function f(x) becomes zero. By substituting -1 into the function, we can verify this claim:

f(-1) = (-1)² - 2(-1) - 3

f(-1) = 1 + 2 - 3

f(-1) = 0

As we can see, when x is -1, the function f(x) equals zero, confirming that -1 is indeed one of the roots of the given function.

Analyzing the first root: The impact of x = -1 on the function f(x) = x² - 2x - 3

Now that we have determined that x = -1 is a root of the function, let's analyze its impact on the overall behavior of f(x) = x² - 2x - 3. By substituting -1 into the function, we can find the corresponding y-value:

f(-1) = (-1)² - 2(-1) - 3

f(-1) = 1 + 2 - 3

f(-1) = 0

This means that when x is -1, the function f(x) evaluates to 0, indicating that the parabolic curve intersects the x-axis at this point. It represents a solution to the equation x² - 2x - 3 = 0 and provides valuable information about the function's behavior.

Discovering the second root: Unveiling the second value of x, unknown at this point

Having found one of the roots as x = -1, we still need to determine the missing number that represents the second root of the function f(x) = x² - 2x - 3. This unknown value of x will also make the function equal to zero. Let's proceed with the investigation to uncover this missing number.

Looking for clues: Examining the given information to find hints about the missing number

To determine the missing number, we must examine the available information and search for clues that can lead us to its discovery. So far, we know that x = -1 is a root of the function f(x) = x² - 2x - 3. By studying the behavior of the parabolic curve and considering the properties of quadratic functions, we can uncover hints that will guide us towards finding the missing number.

Applying the quadratic formula: An approach to calculate the missing value of x

The quadratic formula is a powerful tool used to find the roots of any quadratic function. It states that for a quadratic equation in the form ax² + bx + c = 0, the roots can be found using the formula:

x = (-b ± √(b² - 4ac)) / 2a

By applying this formula to the given function f(x) = x² - 2x - 3, we can calculate the missing value of x and complete the roots of the function.

Factoring the equation: Another method to find the elusive second root

In addition to the quadratic formula, another method to find the missing number is by factoring the equation x² - 2x - 3 = 0. Factoring involves expressing the equation as a product of two binomial expressions. By factoring the quadratic equation, we can easily determine the values of x that make the equation equal to zero, thus revealing the second root.

Determining the missing number: The calculation that leads to finding the second root

By applying the quadratic formula or factoring the equation x² - 2x - 3 = 0, we can determine the missing number that completes the roots of the function f(x) = x² - 2x - 3. By solving the equation, we obtain the value(s) of x that make the function equal to zero:

x = (-b ± √(b² - 4ac)) / 2a

or

x = (factors of the quadratic equation)

By performing the necessary calculations, we will find the missing number that represents the second root of the function, completing our understanding of its behavior and properties.

Conclusion: Revealing the value of x that completes the roots of the function f(x) = x² - 2x - 3

In conclusion, the roots of the function f(x) = x² - 2x - 3 play a significant role in understanding its behavior and characteristics. By determining the first root as x = -1, where f(x) equals zero, we have gained valuable insight into the function's behavior at that particular point. However, there is still a missing number that represents the second root of the function. By employing the quadratic formula or factoring the equation x² - 2x - 3 = 0, we can calculate this missing value of x and complete the roots of the function. The completion of the roots will provide us with a comprehensive understanding of the function f(x) = x² - 2x - 3 and its behavior throughout the coordinate system.

The Roots of the Function F(X) = X^2 – 2x – 3 are Shown

Introduction

In mathematics, a function is a relation that maps each element of a set to exactly one element of another set. The roots of a function are the values of x for which the function equals zero. In this story, we will explore the roots of the function F(x) = x^2 – 2x – 3.

The Function

The function F(x) = x^2 – 2x – 3 can be represented as:

F(x) = x^2 – 2x – 3

Table of Values

Let's construct a table to find the missing number when x = -1 and x = ?. We will substitute different values for x into the function and calculate the corresponding values of F(x).

Finding the Missing Number

To find the missing number, we need to solve the equation F(x) = 0. By substituting different values for x into the function, we can determine the value of x that makes F(x) equal to zero.

Let's begin by substituting x = -1 into the function:

F(-1) = (-1)^2 – 2(-1) – 3

F(-1) = 1 + 2 – 3

F(-1) = 0

As we can see, when x = -1, F(x) equals zero. This means that one of the roots of the function is x = -1.

Next, let's find the missing number by solving the equation F(x) = 0:

x^2 – 2x – 3 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our case, a = 1, b = -2, and c = -3:

x = (-(-2) ± √((-2)^2 - 4(1)(-3))) / (2(1))

x = (2 ± √(4 + 12)) / 2

x = (2 ± √16) / 2

x = (2 ± 4) / 2

Using the plus-minus symbol, we get two possible solutions:

x1 = (2 + 4) / 2 = 6 / 2 = 3

x2 = (2 - 4) / 2 = -2 / 2 = -1

Therefore, the missing number when x = ? is 3. The roots of the function F(x) = x^2 – 2x – 3 are x = -1 and x = 3.

In the first paragraph, we discussed the roots of the function f(x) = x^2 - 2x - 3. By setting the equation equal to zero and employing the quadratic formula, we found that the two roots are x = -1 and x = 3. These values represent the x-coordinates of the points where the function intersects the x-axis. The point (-1, 0) and (3, 0) are known as the x-intercepts or the zeros of the function. They indicate the values of x for which f(x) equals zero.

Now, let's move on to the missing number. We have determined that x = -1 and x = 3 are the roots of the function. However, there seems to be no mention of a specific missing number in the given information. It is essential to clarify what exactly we mean by the missing number. Are we referring to a number that should be added or subtracted from x to make the equation true? Or are we looking for a value that satisfies a specific condition within the context of the function?

In conclusion, while we have explored the roots of the function f(x) = x^2 - 2x - 3, we have encountered a missing number without a specific definition or context. Further clarification is needed to determine the exact nature of the missing number. We hope that this article has provided you with valuable insights into the topic, and we encourage you to continue exploring the fascinating world of functions and their roots.

People Also Ask About the Roots of the Function f(x) = x^2 - 2x - 3

What is the missing number when x = -1 and x = ?

When given the function f(x) = x^2 - 2x - 3, we can find the missing number by solving for the value of x when the function equals zero. This is because the missing number represents the x-value(s) where the function crosses the x-axis and has a root.

To solve for the missing number, we set the function equal to zero:

x^2 - 2x - 3 = 0

To factor or use the quadratic formula to solve this equation, we need to determine the two numbers that multiply to give -3 (the constant term) and add up to -2 (the coefficient of the x-term). The numbers that satisfy these conditions are -3 and +1.

Therefore, the factored form of the equation becomes:

(x - 3)(x + 1) = 0

Setting each factor equal to zero gives us:

x - 3 = 0 or x + 1 = 0

Solving for x in each equation gives us the roots of the function:

x = 3 or x = -1

So, the missing number when x = -1 and x = ? is 3.