Determining the Number of Real Zeros

Oct 13, 2025 by pynkocean.com 98 views

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means 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. In the given function, f(x) = 5x^2 + 5x + 21, the coefficients are a = 5, b = 5, and c = 21.

Determining the Number of Real Zeros

To determine the number of real zeros of a quadratic function, we need to examine the discriminant, which is the expression under the square root in the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is the expression inside the square root, which is b^2 - 4ac. If the discriminant is positive, the quadratic function has two distinct real zeros. If the discriminant is zero, the quadratic function has one real zero. If the discriminant is negative, the quadratic function has no real zeros.

Calculating the Discriminant

To determine the number of real zeros of the given quadratic function, we need to calculate the discriminant. The discriminant is given by:

b^2 - 4ac = (5)^2 - 4(5)(21) = 25 - 420 = -395

Interpreting the Discriminant

Since the discriminant is negative (-395), the quadratic function has no real zeros. This means that the graph of the quadratic function does not intersect the x-axis, and there are no real values of x that satisfy the equation f(x) = 0.

Conclusion

In conclusion, the quadratic function f(x) = 5x^2 + 5x + 21 has no real zeros. This is because the discriminant is negative, indicating that the quadratic function does not intersect the x-axis. Therefore, the correct answer is C. No real zeros.

Real-World Applications

Understanding the number of real zeros of a quadratic function has many real-world applications. For example, in physics, the motion of an object under the influence of gravity can be modeled using a quadratic function. The number of real zeros of the quadratic function can indicate the number of times the object will pass through a certain point or reach a certain height.

Mathematical Implications

The number of real zeros of a quadratic function also has important implications in mathematics. For example, the quadratic formula can be used to solve quadratic equations, and the number of real zeros can indicate the number of solutions to the equation. Additionally, the discriminant can be used to determine the nature of the solutions, whether they are real or complex.

Historical Context

The concept of quadratic functions and their zeros has been studied for centuries. The ancient Greeks, such as Euclid and Archimedes, studied quadratic equations and their solutions. The quadratic formula was first developed by the Indian mathematician Aryabhata in the 5th century. The concept of the discriminant was first introduced by the French mathematician Étienne Bézout in the 18th century.

Future Directions

The study of quadratic functions and their zeros continues to be an active area of research. New techniques and methods are being developed to solve quadratic equations and determine the number of real zeros. The use of computers and numerical methods has made it possible to solve quadratic equations and determine the number of real zeros with high accuracy.

Conclusion

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means 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.

Q: How do I determine the number of real zeros of a quadratic function?

A: To determine the number of real zeros of a quadratic function, you need to examine the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the quadratic function has two distinct real zeros. If the discriminant is zero, the quadratic function has one real zero. If the discriminant is negative, the quadratic function has no real zeros.

Q: What is the discriminant?

A: The discriminant is the expression inside the square root in the quadratic formula, which is b^2 - 4ac. If the discriminant is positive, the quadratic function has two distinct real zeros. If the discriminant is zero, the quadratic function has one real zero. If the discriminant is negative, the quadratic function has no real zeros.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to plug in the values of a, b, and c into the expression b^2 - 4ac. For example, if a = 5, b = 5, and c = 21, the discriminant would be (5)^2 - 4(5)(21) = 25 - 420 = -395.

Q: What does a negative discriminant mean?

A: A negative discriminant means that the quadratic function has no real zeros. This means that the graph of the quadratic function does not intersect the x-axis, and there are no real values of x that satisfy the equation f(x) = 0.

Q: Can a quadratic function have complex zeros?

A: Yes, a quadratic function can have complex zeros. If the discriminant is negative, the quadratic function has complex zeros. Complex zeros are values of x that satisfy the equation f(x) = 0, but are not real numbers.

Q: How do I find the complex zeros of a quadratic function?

A: To find the complex zeros of a quadratic function, you need to use the quadratic formula and take the square root of the discriminant. If the discriminant is negative, the square root will be an imaginary number, which will give you the complex zeros.

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

A: Quadratic functions have many real-world applications, such as modeling the motion of objects under the influence of gravity, determining the maximum height of a projectile, and finding the minimum or maximum value of a function.

Q: Can I use a calculator to find the zeros of a quadratic function?

A: Yes, you can use a calculator to find the zeros of a quadratic function. Most calculators have a built-in quadratic formula function that you can use to find the zeros.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to use a graphing calculator or a computer program. You can also use a table of values to plot the function and find the zeros.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q: Can I use a quadratic function to model a real-world situation?

A: Yes, you can use a quadratic function to model a real-world situation. For example, you can use a quadratic function to model the motion of an object under the influence of gravity, or to find the maximum height of a projectile.

Q: How do I determine the vertex of a quadratic function?

A: To determine the vertex of a quadratic function, you need to use the formula x = -b / 2a. This will give you the x-coordinate of the vertex, and you can then find the y-coordinate by plugging the x-coordinate into the function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on the direction of the parabola.