Determine The Discriminant For The Quadratic Equation $0 = -2x^2 + 3$. Based On The Discriminant Value, How Many Real Number Solutions Does The Equation Have?Discriminant $= B^2 - 4ac$Options:A. 0B. 1C. 2D. 24

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 $0 = -2x^2 + 3$.

The Discriminant Formula

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is used to determine the nature of the solutions of the equation. The formula for the discriminant is:

$$\Delta = b^2 - 4ac$$

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

Applying the Discriminant Formula

In our quadratic equation $0 = -2x^2 + 3$, we have $a = -2$, $b = 0$, and $c = 3$. We can now substitute these values into the discriminant formula:

$$\Delta = b^2 - 4ac$$

$$\Delta = 0^2 - 4(-2)(3)$$

$$\Delta = 0 + 24$$

$$\Delta = 24$$

Interpreting the Discriminant Value

The discriminant value is 24. This value tells us about the nature of the solutions of the quadratic equation.

Real Number Solutions

If the discriminant value is positive, the quadratic equation has two distinct real number solutions. If the discriminant value is zero, the quadratic equation has one real number solution. If the discriminant value is negative, the quadratic equation has no real number solutions.

Conclusion

Based on the discriminant value of 24, we can conclude that the quadratic equation $0 = -2x^2 + 3$ has two distinct real number solutions.

Answer

The correct answer is:

• C. 2

Additional Information

The discriminant value can also be used to determine the nature of the solutions of the quadratic equation. If the discriminant value is a perfect square, the solutions are rational numbers. If the discriminant value is not a perfect square, the solutions are irrational numbers.

Example Use Case

Suppose we have a quadratic equation $0 = x^2 + 4x + 4$. We can calculate the discriminant value as follows:

$$\Delta = b^2 - 4ac$$

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

$$\Delta = 16 - 16$$

$$\Delta = 0$$

Since the discriminant value is zero, we can conclude that the quadratic equation has one real number solution.

Conclusion

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 used to determine the nature of the solutions of the equation.

Q: How is the discriminant calculated?

A: The discriminant is calculated using the formula:

$$\Delta = b^2 - 4ac$$

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

Q: What does the discriminant value tell us about the solutions of the quadratic equation?

A: The discriminant value tells us about the nature of the solutions of the quadratic equation. If the discriminant value is positive, the quadratic equation has two distinct real number solutions. If the discriminant value is zero, the quadratic equation has one real number solution. If the discriminant value is negative, the quadratic equation has no real number solutions.

Q: Can the discriminant value be used to determine the nature of the solutions of the quadratic equation?

A: Yes, the discriminant value can be used to determine the nature of the solutions of the quadratic equation. If the discriminant value is a perfect square, the solutions are rational numbers. If the discriminant value is not a perfect square, the solutions are irrational numbers.

Q: How is the discriminant used in real-world applications?

A: The discriminant is used in various real-world applications, such as:

• Physics: The discriminant is used to determine the nature of the solutions of equations that describe the motion of objects.
• Engineering: The discriminant is used to determine the stability of structures and systems.
• Computer Science: The discriminant is used in algorithms for solving quadratic equations.

Q: Can the discriminant be used to solve quadratic equations?

A: Yes, the discriminant can be used to solve quadratic equations. If the discriminant value is positive, the quadratic equation has two distinct real number solutions. If the discriminant value is zero, the quadratic equation has one real number solution. If the discriminant value is negative, the quadratic equation has no real number solutions.

Q: What are some common mistakes to avoid when calculating the discriminant?

A: Some common mistakes to avoid when calculating the discriminant include:

• Incorrectly identifying the coefficients: Make sure to identify the correct coefficients of the quadratic equation.
• Incorrectly applying the formula: Make sure to apply the formula correctly.
• Not checking the discriminant value: Make sure to check the discriminant value to determine the nature of the solutions of the quadratic equation.

Q: Can the discriminant be used to determine the number of solutions of a quadratic equation?

A: Yes, the discriminant can be used to determine the number of solutions of a quadratic equation. If the discriminant value is positive, the quadratic equation has two distinct real number solutions. If the discriminant value is zero, the quadratic equation has one real number solution. If the discriminant value is negative, the quadratic equation has no real number solutions.

Q: What are some real-world examples of using the discriminant?

A: Some real-world examples of using the discriminant include:

• Designing a bridge: The discriminant is used to determine the stability of the bridge.
• Optimizing a system: The discriminant is used to determine the optimal solution of a system.
• Solving a quadratic equation: The discriminant is used to determine the number of solutions of a quadratic equation.

Conclusion

In conclusion, the discriminant is a useful tool for determining the nature of the solutions of a quadratic equation. By calculating the discriminant value, we can determine whether the equation has one or two real number solutions, or no real number solutions at all.