The Quadratic Function in Question

Oct 13, 2025 by talynt.org 29 views

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their behavior is crucial in various fields such as physics, engineering, and economics. One of the key aspects of quadratic functions is the number of real-number solutions they have. In this article, we will explore how to determine the number of real-number solutions of a quadratic function using the discriminant.

What is a Quadratic Function?

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:

y = ax^2 + bx + c

where a, b, and c are constants, and x is the variable.

The Quadratic Function in Question

The quadratic function we will be working with is:

y = -5x^2 + 6x + 4

This function has a negative leading coefficient (a = -5), which means it opens downward.

The Discriminant

The discriminant is a value that can be calculated from the coefficients of a quadratic function. It is used to determine the number of real-number solutions of the function. The discriminant is calculated using the following formula:

b^2 - 4ac

where a, b, and c are the coefficients of the quadratic function.

Calculating the Discriminant

To calculate the discriminant of the quadratic function y = -5x^2 + 6x + 4, we need to substitute the values of a, b, and c into the formula.

a = -5 b = 6 c = 4

Now, let's calculate the discriminant:

b^2 - 4ac = (6)^2 - 4(-5)(4) = 36 + 80 = 116

Interpreting the Discriminant

The discriminant is a positive value (116), which means the quadratic function has two real-number solutions.

Why is the Discriminant Positive?

The discriminant is positive because the expression b^2 - 4ac is positive. This is because the square of b (36) is greater than 4 times the product of a and c (80).

What Does a Positive Discriminant Mean?

A positive discriminant means that the quadratic function has two real-number solutions. This is because the graph of the function will intersect the x-axis at two points.

Conclusion

In conclusion, the quadratic function y = -5x^2 + 6x + 4 has two real-number solutions. This is determined by the discriminant, which is a positive value (116). The discriminant is a useful tool for determining the number of real-number solutions of a quadratic function.

The Final Answer

The final answer is:

A. Two real-number solutions

Additional Information

The discriminant can also be used to determine the nature of the solutions of a quadratic function. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.
The discriminant can also be used to determine the number of real-number solutions of a quadratic function in the form of y = ax^2 + bx + c. If the discriminant is positive, the function has two real-number solutions. If the discriminant is zero, the function has one real-number solution. If the discriminant is negative, the function has no real-number solutions.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Discriminant" by Wolfram MathWorld
[3] "Quadratic Equations" by Khan Academy
Quadratic Function Solutions: A Q&A Guide =============================================

Introduction

In our previous article, we explored how to determine the number of real-number solutions of a quadratic function using the discriminant. In this article, we will answer some frequently asked questions about quadratic function solutions.

Q: What is the discriminant, and how is it used to determine the number of real-number solutions of a quadratic function?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic function. It is used to determine the number of real-number solutions of the function. The discriminant is calculated using the formula:

b^2 - 4ac

where a, b, and c are the coefficients of the quadratic function.

Q: What does a positive discriminant mean?

A: A positive discriminant means that the quadratic function has two real-number solutions. This is because the graph of the function will intersect the x-axis at two points.

Q: What does a zero discriminant mean?

A: A zero discriminant means that the quadratic function has one real-number solution. This is because the graph of the function will intersect the x-axis at one point.

Q: What does a negative discriminant mean?

A: A negative discriminant means that the quadratic function has no real-number solutions. This is because the graph of the function will not intersect the x-axis.

Q: Can the discriminant be used to determine the nature of the solutions of a quadratic function?

A: Yes, the discriminant can be used to determine the nature of the solutions of a quadratic function. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q: How is the discriminant used in the quadratic formula?

A: The discriminant is used in the quadratic formula to determine the nature of the solutions of a quadratic function. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and equal. If the discriminant is negative, the solutions are complex.

Q: Can the discriminant be used to determine the number of real-number solutions of a quadratic function in the form of y = ax^2 + bx + c?

A: Yes, the discriminant can be used to determine the number of real-number solutions of a quadratic function in the form of y = ax^2 + bx + c. If the discriminant is positive, the function has two real-number solutions. If the discriminant is zero, the function has one real-number solution. If the discriminant is negative, the function has no real-number solutions.

Q: What are some common mistakes to avoid when using the discriminant to determine the number of real-number solutions of a quadratic function?

A: Some common mistakes to avoid when using the discriminant to determine the number of real-number solutions of a quadratic function include:

Not calculating the discriminant correctly
Not understanding the meaning of the discriminant
Not using the correct formula for the discriminant
Not considering the nature of the solutions of the quadratic function

Conclusion

In conclusion, the discriminant is a powerful tool for determining the number of real-number solutions of a quadratic function. By understanding how to calculate and interpret the discriminant, you can solve quadratic equations and determine the nature of the solutions.

Additional Information

The discriminant can also be used to determine the number of real-number solutions of a quadratic function in the form of y = ax^2 + bx + c.
The discriminant can also be used to determine the nature of the solutions of a quadratic function.
The quadratic formula can be used to determine the solutions of a quadratic function.

References

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