What Is The Discriminant, And How Many Solutions Does This Quadratic Equation Have?${-4x^2 - 8x - 8 = -4}$A. 128, 2 Solutions B. 128, 1 Solution C. 0, 1 Solution

Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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.

The Discriminant: A Key to Understanding Quadratic Equations

The discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions. The discriminant is denoted by the symbol Δ (delta) or D, and it is calculated using the formula Δ = b^2 - 4ac. The discriminant is a key to understanding the number of solutions a quadratic equation has.

How to Calculate the Discriminant

To calculate the discriminant, we need to substitute the values of a, b, and c into the formula Δ = b^2 - 4ac. Let's consider the given quadratic equation -4x^2 - 8x - 8 = -4. We can rewrite this equation as -4x^2 - 8x - 4 = 0. Now, we can identify the values of a, b, and c: a = -4, b = -8, and c = -4.

Calculating the Discriminant for the Given Quadratic Equation

Now, let's calculate the discriminant using the formula Δ = b^2 - 4ac. Substituting the values of a, b, and c, we get:

Δ = (-8)^2 - 4(-4)(-4) Δ = 64 - 64 Δ = 0

Understanding the Significance of the Discriminant

The discriminant is a key to understanding 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.

Determining the Number of Solutions Based on the Discriminant

Based on the value of the discriminant, we can determine the number of solutions the quadratic equation has. In this case, the discriminant is zero, which means the equation has one real solution.

Conclusion

In conclusion, the discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula Δ = b^2 - 4ac. Based on the value of the discriminant, we can determine the number of solutions the quadratic equation has. In this case, the discriminant is zero, which means the equation has one real solution.

Frequently Asked Questions

• What is the discriminant? The discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions.
• How is the discriminant calculated? The discriminant is calculated using the formula Δ = b^2 - 4ac.
• What does the value of the discriminant indicate? The value of the discriminant indicates the number of solutions the quadratic equation has. 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.

Final Answer

The final answer is C. 0, 1 solution.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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.

Understanding Quadratic Equation Solutions

Quadratic equations can have one, two, or no real solutions. The number of solutions depends on the value of the discriminant, which is calculated using the formula Δ = b^2 - 4ac. In this article, we will explore the different types of solutions and how to determine the number of solutions based on the discriminant.

Q&A: Quadratic Equation Solutions

Q: What is the discriminant, and how is it calculated?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula Δ = b^2 - 4ac.

Q: What does the value of the discriminant indicate?

A: The value of the discriminant indicates the number of solutions the quadratic equation has. 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 determine the number of solutions based on the discriminant?

A: To determine the number of solutions, you need to calculate the discriminant using the formula Δ = b^2 - 4ac. 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: What is the significance of the discriminant in quadratic equations?

A: The discriminant is a key to understanding the nature of the solutions of a quadratic equation. It helps us determine the number of solutions and whether the solutions are real or complex.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it can have complex solutions.

Q: How do I find the solutions of a quadratic equation?

A: To find the solutions of a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula gives you the two solutions of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives you the solutions of a quadratic equation. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: Can a quadratic equation have one solution?

A: Yes, a quadratic equation can have one solution. If the discriminant is zero, the equation has one real solution.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. If the discriminant is negative, the equation has no real solutions.

Conclusion

In conclusion, the discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula Δ = b^2 - 4ac. Based on the value of the discriminant, we can determine the number of solutions the quadratic equation has. We hope this article has provided you with a comprehensive understanding of quadratic equation solutions.

Frequently Asked Questions

• What is the discriminant? The discriminant is a value that can be calculated from the coefficients of a quadratic equation, and it plays a crucial role in determining the nature of the solutions.
• How is the discriminant calculated? The discriminant is calculated using the formula Δ = b^2 - 4ac.
• What does the value of the discriminant indicate? The value of the discriminant indicates the number of solutions the quadratic equation has. 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.

Final Answer

The final answer is C. 0, 1 solution.