Consider The Function F ( X ) = X 2 + 2 X − 15 F(x)=x^2+2x-15 F ( X ) = X 2 + 2 X − 15 . What Are The X X X -intercepts Of The Function?- Left-most X X X -intercept: ( □ \square □ , 0)- Right-most X X X -intercept: ( □ \square □ , 0)

Introduction

In mathematics, the x-intercepts of a function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or solutions of the function. In this article, we will explore how to find the x-intercepts of a quadratic function, specifically the function $f(x)=x^2+2x-15$.

What are x-Intercepts?

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. At these points, the value of the function is equal to zero. In other words, the x-intercepts are the solutions to the equation $f(x)=0$. The x-intercepts are important because they provide valuable information about the behavior of the function.

Quadratic Functions

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. The function $f(x)=x^2+2x-15$ is a specific example of a quadratic function.

Finding the x-Intercepts

To find the x-intercepts of a quadratic function, we need to solve the equation $f(x)=0$. In other words, we need to find the values of x that make the function equal to zero. We can do this by factoring the quadratic expression or by using the quadratic formula.

Factoring the Quadratic Expression

The quadratic expression $x^2+2x-15$ can be factored as $(x+5)(x-3)$. To factor the expression, we need to find two numbers whose product is equal to the constant term (-15) and whose sum is equal to the coefficient of the linear term (2). In this case, the numbers are 5 and -3.

Using the Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is given by:

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

In this case, the values of $a$, $b$, and $c$ are 1, 2, and -15, respectively. Plugging these values into the formula, we get:

$x=\frac{-2\pm\sqrt{2^2-4(1)(-15)}}{2(1)}$ $x=\frac{-2\pm\sqrt{4+60}}{2}$ $x=\frac{-2\pm\sqrt{64}}{2}$ $x=\frac{-2\pm8}{2}$

Simplifying the expression, we get two possible values for x:

$x_1=\frac{-2+8}{2}=3$ $x_2=\frac{-2-8}{2}=-5$

Conclusion

In this article, we have explored how to find the x-intercepts of a quadratic function, specifically the function $f(x)=x^2+2x-15$. We have used two methods to find the x-intercepts: factoring the quadratic expression and using the quadratic formula. The x-intercepts are the points where the graph of the function crosses the x-axis, and they provide valuable information about the behavior of the function.

Left-most x-Intercept

The left-most x-intercept is the point where the graph of the function crosses the x-axis from the left. In this case, the left-most x-intercept is (-5, 0).

Right-most x-Intercept

The right-most x-intercept is the point where the graph of the function crosses the x-axis from the right. In this case, the right-most x-intercept is (3, 0).

Final Answer

Introduction

In our previous article, we explored how to find the x-intercepts of a quadratic function, specifically the function $f(x)=x^2+2x-15$. In this article, we will answer some common questions related to quadratic function x-intercepts.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or solutions of the function.

Q: How do I find the x-intercepts of a quadratic function?

A: There are two common methods to find the x-intercepts of a quadratic function: factoring the quadratic expression and using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is given by:

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

Q: How do I use the quadratic formula to find the x-intercepts?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. The values of $a$, $b$, and $c$ are the coefficients of the quadratic expression.

Q: What are the x-intercepts of the function $f(x)=x^2+2x-15$?

A: The x-intercepts of the function $f(x)=x^2+2x-15$ are (-5, 0) and (3, 0).

Q: How do I determine the left-most and right-most x-intercepts?

A: The left-most x-intercept is the point where the graph of the function crosses the x-axis from the left. The right-most x-intercept is the point where the graph of the function crosses the x-axis from the right.

Q: What is the significance of the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function provide valuable information about the behavior of the function. They can be used to determine the maximum or minimum value of the function, as well as the intervals where the function is increasing or decreasing.

Q: Can I use the x-intercepts to graph a quadratic function?

A: Yes, you can use the x-intercepts to graph a quadratic function. The x-intercepts are the points where the graph of the function crosses the x-axis, and they can be used to draw the graph of the function.

Q: What are some common mistakes to avoid when finding the x-intercepts of a quadratic function?

A: Some common mistakes to avoid when finding the x-intercepts of a quadratic function include:

• Not factoring the quadratic expression correctly
• Not using the correct values of $a$, $b$, and $c$ in the quadratic formula
• Not simplifying the expression correctly
• Not checking for extraneous solutions

Conclusion

In this article, we have answered some common questions related to quadratic function x-intercepts. We have discussed the significance of the x-intercepts, how to find them using factoring and the quadratic formula, and how to determine the left-most and right-most x-intercepts. We have also provided some tips on how to avoid common mistakes when finding the x-intercepts of a quadratic function.