What Is The Discriminant Of The Quadratic Equation $-x^2 + 3x + 8 = 0$?A. 41 B. 23 C. -23 D. -41

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this case, we have the quadratic equation $-x^2 + 3x + 8 = 0$.

The Discriminant: A Key to Solving Quadratic Equations

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the letter $D$ or $\Delta$, and it is calculated using the formula $D = b^2 - 4ac$. The discriminant is a crucial component in solving quadratic equations, as it determines the nature of the solutions.

Calculating the Discriminant

To calculate the discriminant of the quadratic equation $-x^2 + 3x + 8 = 0$, we need to identify the values of $a$, $b$, and $c$. In this case, $a = -1$, $b = 3$, and $c = 8$. Now, we can plug these values into the formula for the discriminant:

$$D = b^2 - 4ac$$

$$D = (3)^2 - 4(-1)(8)$$

$$D = 9 + 32$$

$$D = 41$$

Interpreting the Discriminant

The discriminant $D = 41$ indicates that the quadratic equation $-x^2 + 3x + 8 = 0$ has two distinct real solutions. This is because the discriminant is positive, which means the quadratic equation has two real and distinct roots.

Conclusion

In conclusion, the discriminant of the quadratic equation $-x^2 + 3x + 8 = 0$ is $41$. This value indicates that the equation has two distinct real solutions. The discriminant is a crucial component in solving quadratic equations, and it plays a vital role in determining the nature of the solutions.

Frequently Asked Questions

• What is the discriminant of a quadratic equation? The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the letter $D$ or $\Delta$, and it is calculated using the formula $D = b^2 - 4ac$.
• What is the significance of the discriminant in solving quadratic equations? The discriminant determines the nature of the solutions of a quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Real-World Applications of the Discriminant

The discriminant has numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design and optimize systems. In economics, the discriminant is used to model and analyze economic systems.

Conclusion

In conclusion, the discriminant of the quadratic equation $-x^2 + 3x + 8 = 0$ is $41$. This value indicates that the equation has two distinct real solutions. The discriminant is a crucial component in solving quadratic equations, and it plays a vital role in determining the nature of the solutions.

Understanding the Quadratic Equation Discriminant

The quadratic equation discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is denoted by the letter $D$ or $\Delta$, and it is calculated using the formula $D = b^2 - 4ac$. The discriminant is a crucial component in solving quadratic equations, and it determines the nature of the solutions.

Frequently Asked Questions

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. It is denoted by the letter $D$ or $\Delta$, and it is calculated using the formula $D = b^2 - 4ac$.

Q: What is the significance of the discriminant in solving quadratic equations?

A: The discriminant determines the nature of the solutions of a quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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

A: To calculate the discriminant of a quadratic equation, you need to identify the values of $a$, $b$, and $c$. Then, you can plug these values into the formula for the discriminant: $D = b^2 - 4ac$.

Q: What is the difference between a positive, zero, and negative discriminant?

A: A positive discriminant indicates that the quadratic equation has two distinct real solutions. A zero discriminant indicates that the quadratic equation has one real solution. A negative discriminant indicates that the quadratic equation has no real solutions.

Q: Can the discriminant be used to determine the stability of a system?

A: Yes, the discriminant can be used to determine the stability of a system. If the discriminant is positive, the system is stable. If the discriminant is negative, the system is unstable.

Q: Can the discriminant be used to design and optimize systems?

A: Yes, the discriminant can be used to design and optimize systems. By analyzing the discriminant, you can determine the optimal values of the system's parameters.

Q: Can the discriminant be used to model and analyze economic systems?

A: Yes, the discriminant can be used to model and analyze economic systems. By analyzing the discriminant, you can determine the optimal values of the economic system's parameters.

Real-World Applications of the Discriminant

The discriminant has numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design and optimize systems. In economics, the discriminant is used to model and analyze economic systems.

Conclusion

In conclusion, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the letter $D$ or $\Delta$, and it is calculated using the formula $D = b^2 - 4ac$. The discriminant determines the nature of the solutions of a quadratic equation, and it has numerous real-world applications in various fields.

Additional Resources

• Quadratic Equation Formula: $ax^2 + bx + c = 0$
• Discriminant Formula: $D = b^2 - 4ac$
• Quadratic Equation Solutions: $x = \frac{-b \pm \sqrt{D}}{2a}$

Final Thoughts

The discriminant is a crucial component in solving quadratic equations, and it determines the nature of the solutions. By understanding the discriminant, you can solve quadratic equations and apply the concepts to real-world problems.