What Is The Discriminant Of $9x^2 + 2 = 10x$?A. -356 B. -172 C. 28 D. 72
Introduction to the Discriminant
The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the solutions. In this article, we will explore the concept of the discriminant and how to calculate it for a given quadratic equation.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and a cannot be zero.
The Discriminant Formula
The discriminant of a quadratic equation is given by the formula:
Δ = b^2 - 4ac
where Δ is the discriminant, and a, b, and c are the coefficients of the quadratic equation.
Calculating the Discriminant
To calculate the discriminant, we need to identify the coefficients a, b, and c in the quadratic equation. In the given equation $9x^2 + 2 = 10x$, we can rewrite it in the standard form as:
9x^2 - 10x + 2 = 0
Comparing this with the general form of a quadratic equation, we can see that a = 9, b = -10, and c = 2.
Applying the Discriminant Formula
Now that we have identified the coefficients, we can apply the discriminant formula:
Δ = b^2 - 4ac Δ = (-10)^2 - 4(9)(2) Δ = 100 - 72 Δ = 28
Conclusion
The discriminant of the quadratic equation $9x^2 + 2 = 10x$ is 28. This value provides information about the nature of the solutions, which can be used to determine the number and type of solutions the equation has.
Understanding the Nature of the Solutions
The discriminant can be used to determine the nature of the solutions of a quadratic equation. If the discriminant is:
- Positive (Δ > 0), the equation has two distinct real solutions.
- Zero (Δ = 0), the equation has one real solution (or two equal real solutions).
- Negative (Δ < 0), the equation has no real solutions.
In this case, since the discriminant is positive (Δ = 28), the equation has two distinct real solutions.
Final Answer
The final answer to the question "What is the discriminant of $9x^2 + 2 = 10x$?" is:
C. 28
Introduction
The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the solutions. In this article, we will answer some frequently asked questions about the discriminant of a quadratic equation.
Q: What is the discriminant of a quadratic equation?
A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the solutions. It is given by the formula:
Δ = b^2 - 4ac
where Δ is the discriminant, and a, b, and c are the coefficients of the quadratic equation.
Q: How do I calculate the discriminant of a quadratic equation?
A: To calculate the discriminant, you need to identify the coefficients a, b, and c in the quadratic equation. Then, you can apply the discriminant formula:
Δ = b^2 - 4ac
Q: What does the discriminant tell me about the solutions of a quadratic equation?
A: The discriminant can be used to determine the nature of the solutions of a quadratic equation. If the discriminant is:
- Positive (Δ > 0), the equation has two distinct real solutions.
- Zero (Δ = 0), the equation has one real solution (or two equal real solutions).
- Negative (Δ < 0), the equation has no real solutions.
Q: How do I determine the number of solutions of a quadratic equation using the discriminant?
A: To determine the number of solutions of a quadratic equation using the discriminant, you can follow these steps:
- Calculate the discriminant using the formula Δ = b^2 - 4ac.
- If the discriminant is positive (Δ > 0), the equation has two distinct real solutions.
- If the discriminant is zero (Δ = 0), the equation has one real solution (or two equal real solutions).
- If the discriminant is negative (Δ < 0), the equation has no real solutions.
Q: What is the significance of the discriminant in real-world applications?
A: The discriminant has significant applications in various fields, including:
- Physics: The discriminant is used to determine the stability of a system.
- Engineering: The discriminant is used to design and optimize systems.
- Economics: The discriminant is used to model and analyze economic systems.
Q: Can I use the discriminant to solve quadratic equations?
A: Yes, you can use the discriminant to solve quadratic equations. However, the discriminant only provides information about the nature of the solutions, not the actual solutions. To find the actual solutions, you need to use other methods, such as factoring or the quadratic formula.
Q: What are some common mistakes to avoid when calculating the discriminant?
A: Some common mistakes to avoid when calculating the discriminant include:
- Not identifying the correct coefficients a, b, and c.
- Not applying the correct formula Δ = b^2 - 4ac.
- Not checking the sign of the discriminant.
Q: Can I use a calculator to calculate the discriminant?
A: Yes, you can use a calculator to calculate the discriminant. Most calculators have a built-in function to calculate the discriminant.
Q: What is the relationship between the discriminant and the quadratic formula?
A: The discriminant is used in the quadratic formula to determine the nature of the solutions. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
The discriminant is the value inside the square root, Δ = b^2 - 4ac.
Q: Can I use the discriminant to determine the type of solutions of a quadratic equation?
A: Yes, you can use the discriminant to determine the type of solutions of a quadratic equation. If the discriminant is:
- Positive (Δ > 0), the equation has two distinct real solutions.
- Zero (Δ = 0), the equation has one real solution (or two equal real solutions).
- Negative (Δ < 0), the equation has no real solutions.
Q: What is the final answer to the question "What is the discriminant of $9x^2 + 2 = 10x$"?
A: The final answer to the question "What is the discriminant of $9x^2 + 2 = 10x$" is:
C. 28