$f(x) = 25x^2 - 10x + 1$1. What Is The Value Of The Discriminant Of F F F ? $\square$2. How Many X X X -intercepts Does The Graph Of F F F Have? □ \square □

Understanding the Quadratic Function $f(x) = 25x^2 - 10x + 1$

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on the quadratic function $f(x) = 25x^2 - 10x + 1$ and explore its properties, including the value of the discriminant and the number of $x$-intercepts.

The Discriminant of a Quadratic Function

The discriminant of a quadratic function is a value that can be calculated from the coefficients of the function and is used to determine the nature of the function's roots. The discriminant is denoted by the symbol $\Delta$ or $D$ and is calculated using the formula $\Delta = b^2 - 4ac$. In the case of the quadratic function $f(x) = 25x^2 - 10x + 1$, the coefficients are $a = 25$, $b = -10$, and $c = 1$.

Calculating the Discriminant

To calculate the discriminant, we substitute the values of $a$, $b$, and $c$ into the formula $\Delta = b^2 - 4ac$. This gives us:

$$\Delta = (-10)^2 - 4(25)(1)$$

Simplifying the expression, we get:

$$\Delta = 100 - 100$$

$$\Delta = 0$$

Therefore, the value of the discriminant of the quadratic function $f(x) = 25x^2 - 10x + 1$ is 0.

The Number of $x$-Intercepts

The $x$-intercepts of a quadratic function are the points where the function intersects the $x$-axis. In other words, they are the solutions to the equation $f(x) = 0$. The number of $x$-intercepts of a quadratic function can be determined by the value of the discriminant.

Understanding the Relationship Between the Discriminant and the Number of $x$-Intercepts

If the discriminant is positive, the quadratic function has two distinct $x$-intercepts. If the discriminant is zero, the quadratic function has one $x$-intercept. If the discriminant is negative, the quadratic function has no $x$-intercepts.

In the case of the quadratic function $f(x) = 25x^2 - 10x + 1$, the discriminant is 0. Therefore, the function has one $x$-intercept.

Finding the $x$-Intercept

To find the $x$-intercept of the quadratic function $f(x) = 25x^2 - 10x + 1$, we set the function equal to 0 and solve for $x$. This gives us:

$$25x^2 - 10x + 1 = 0$$

We can solve this equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$

Substituting the values of $a$, $b$, and $\Delta$, we get:

$$x = \frac{-(-10) \pm \sqrt{0}}{2(25)}$$

Simplifying the expression, we get:

$$x = \frac{10}{50}$$

$$x = \frac{1}{5}$$

Therefore, the $x$-intercept of the quadratic function $f(x) = 25x^2 - 10x + 1$ is $x = \frac{1}{5}$.

Conclusion

In conclusion, the quadratic function $f(x) = 25x^2 - 10x + 1$ has a discriminant of 0, which means it has one $x$-intercept. The $x$-intercept of the function is $x = \frac{1}{5}$. Understanding the properties of quadratic functions, including the discriminant and the number of $x$-intercepts, is essential in mathematics and has numerous applications in various fields.

Frequently Asked Questions

• What is the discriminant of a quadratic function? The discriminant of a quadratic function is a value that can be calculated from the coefficients of the function and is used to determine the nature of the function's roots.
• How many $x$-intercepts does the graph of a quadratic function have? The number of $x$-intercepts of a quadratic function can be determined by the value of the discriminant. If the discriminant is positive, the function has two distinct $x$-intercepts. If the discriminant is zero, the function has one $x$-intercept. If the discriminant is negative, the function has no $x$-intercepts.
• How do I find the $x$-intercept of a quadratic function? To find the $x$-intercept of a quadratic function, you can set the function equal to 0 and solve for $x$ using the quadratic formula.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Discriminant" by Wolfram MathWorld
• [3] "Quadratic Formula" by Khan Academy
  Quadratic Function Q&A

In this article, we will answer some frequently asked questions about quadratic functions, including the discriminant, the number of $x$-intercepts, and how to find the $x$-intercept.

Q: What is the discriminant of a quadratic function?

A: The discriminant of a quadratic function is a value that can be calculated from the coefficients of the function and is used to determine the nature of the function's roots. It is calculated using the formula $\Delta = b^2 - 4ac$.

Q: How do I calculate the discriminant of a quadratic function?

A: To calculate the discriminant of a quadratic function, you need to substitute the values of $a$, $b$, and $c$ into the formula $\Delta = b^2 - 4ac$. For example, if the quadratic function is $f(x) = 25x^2 - 10x + 1$, the discriminant would be calculated as follows:

$$\Delta = (-10)^2 - 4(25)(1)$$

Simplifying the expression, we get:

$$\Delta = 100 - 100$$

$$\Delta = 0$$

Q: What does the discriminant tell me about the quadratic function?

A: The discriminant tells you about the nature of the function's roots. If the discriminant is positive, the function has two distinct $x$-intercepts. If the discriminant is zero, the function has one $x$-intercept. If the discriminant is negative, the function has no $x$-intercepts.

Q: How many $x$-intercepts does the graph of a quadratic function have?

A: The number of $x$-intercepts of a quadratic function can be determined by the value of the discriminant. If the discriminant is positive, the function has two distinct $x$-intercepts. If the discriminant is zero, the function has one $x$-intercept. If the discriminant is negative, the function has no $x$-intercepts.

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

A: To find the $x$-intercept of a quadratic function, you can set the function equal to 0 and solve for $x$ using the quadratic formula:

$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$

For example, if the quadratic function is $f(x) = 25x^2 - 10x + 1$, the $x$-intercept would be calculated as follows:

$$x = \frac{-(-10) \pm \sqrt{0}}{2(25)}$$

Simplifying the expression, we get:

$$x = \frac{10}{50}$$

$$x = \frac{1}{5}$$

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to substitute the values of $a$, $b$, and $c$ into the formula and simplify the expression. For example, if the quadratic equation is $25x^2 - 10x + 1 = 0$, the solution would be calculated as follows:

$$x = \frac{-(-10) \pm \sqrt{0}}{2(25)}$$

Simplifying the expression, we get:

$$x = \frac{10}{50}$$

$$x = \frac{1}{5}$$

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

• Not checking the discriminant before solving the quadratic equation
• Not using the correct formula to solve the quadratic equation
• Not simplifying the expression correctly
• Not checking the solution for extraneous solutions

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including:

• Modeling the motion of objects under the influence of gravity
• Describing the shape of a parabola
• Finding the maximum or minimum value of a function
• Solving problems in physics, engineering, and economics

Conclusion

In conclusion, quadratic functions are an important topic in mathematics, and understanding their properties and applications is essential for success in various fields. By mastering the concepts of discriminant, $x$-intercepts, and the quadratic formula, you will be able to solve a wide range of problems and apply quadratic functions to real-world situations.

Frequently Asked Questions

• What is the discriminant of a quadratic function? The discriminant of a quadratic function is a value that can be calculated from the coefficients of the function and is used to determine the nature of the function's roots.
• How do I calculate the discriminant of a quadratic function? To calculate the discriminant of a quadratic function, you need to substitute the values of $a$, $b$, and $c$ into the formula $\Delta = b^2 - 4ac$.
• What does the discriminant tell me about the quadratic function? The discriminant tells you about the nature of the function's roots. If the discriminant is positive, the function has two distinct $x$-intercepts. If the discriminant is zero, the function has one $x$-intercept. If the discriminant is negative, the function has no $x$-intercepts.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Discriminant" by Wolfram MathWorld
• [3] "Quadratic Formula" by Khan Academy