For The Following Quadratic Equation, Find The Discriminant.$-5x^2 - 87x - 391 = 3x - 6$
Introduction
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on finding the discriminant of a quadratic equation, which is a crucial step in solving quadratic equations.
What is the Discriminant?
The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol Δ (delta) or b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant is used to determine the nature of the roots of the quadratic equation.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. The discriminant is the value inside the square root, which is b^2 - 4ac.
Finding the Discriminant
To find the discriminant, we need to identify the coefficients of the quadratic equation. In this case, the quadratic equation is -5x^2 - 87x - 391 = 3x - 6. We can rewrite this equation as -5x^2 - 87x - 391 + 3x - 6 = 0.
Step 1: Identify the Coefficients
The coefficients of the quadratic equation are a = -5, b = -87 + 3 = -84, and c = -391 - 6 = -397.
Step 2: Calculate the Discriminant
Now that we have identified the coefficients, we can calculate the discriminant using the formula b^2 - 4ac.
Δ = b^2 - 4ac = (-84)^2 - 4(-5)(-397) = 7056 - 7960 = -904
Conclusion
In this article, we have discussed the concept of the discriminant and how to find it for a quadratic equation. We have used the quadratic formula to solve the given quadratic equation and calculated the discriminant using the formula b^2 - 4ac. The discriminant is a crucial value that determines the nature of the roots of the quadratic equation.
Nature of the Roots
The nature of the roots of a quadratic equation can be determined by the discriminant. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.
Example
Let's consider an example to illustrate the concept of the discriminant. Suppose we have a quadratic equation x^2 + 5x + 6 = 0. We can calculate the discriminant using the formula b^2 - 4ac.
Δ = b^2 - 4ac = (5)^2 - 4(1)(6) = 25 - 24 = 1
Since the discriminant is positive, the equation has two distinct real roots.
Real-World Applications
The concept of the discriminant has numerous real-world applications. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design stable structures. In economics, the discriminant is used to model the behavior of economic systems.
Conclusion
In conclusion, the discriminant is a crucial value that determines the nature of the roots of a quadratic equation. It is calculated using the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant has numerous real-world applications and is an essential tool in various fields such as physics, engineering, and economics.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Discriminant" by Wolfram MathWorld
- [3] "Quadratic Formula" by Khan Academy
Glossary
- Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
- Discriminant: A value that can be calculated from the coefficients of a quadratic equation, which determines the nature of the roots of the equation.
- Quadratic Formula: A powerful tool for solving quadratic equations, which is given by x = (-b ± √(b^2 - 4ac)) / 2a.
Quadratic Equations: A Q&A Guide =====================================
Introduction
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed the concept of the discriminant and how to find it for a quadratic equation. In this article, we will answer some frequently asked questions about quadratic equations and the discriminant.
Q&A
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
Q: What is the discriminant?
A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation, which determines the nature of the roots of the equation. It is denoted by the symbol Δ (delta) or b^2 - 4ac.
Q: How do I find the discriminant?
A: To find the discriminant, you need to identify the coefficients of the quadratic equation and use the formula b^2 - 4ac.
Q: What does the discriminant tell me about the roots of the equation?
A: The discriminant tells you whether the equation has two distinct real roots, one real root, or no real roots. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.
Q: How do I use the quadratic formula to solve a quadratic equation?
A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a. To use the quadratic formula, you need to identify the coefficients of the quadratic equation and plug them into the formula.
Q: What are some real-world applications of quadratic equations and the discriminant?
A: Quadratic equations and the discriminant have numerous real-world applications. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design stable structures. In economics, the discriminant is used to model the behavior of economic systems.
Q: Can you give me an example of how to find the discriminant?
A: Let's consider an example. Suppose we have a quadratic equation x^2 + 5x + 6 = 0. We can calculate the discriminant using the formula b^2 - 4ac.
Δ = b^2 - 4ac = (5)^2 - 4(1)(6) = 25 - 24 = 1
Since the discriminant is positive, the equation has two distinct real roots.
Q: What if the discriminant is negative?
A: If the discriminant is negative, the equation has no real roots. This means that the equation has complex roots.
Q: Can you give me an example of how to use the quadratic formula to solve a quadratic equation?
A: Let's consider an example. Suppose we have a quadratic equation x^2 + 5x + 6 = 0. We can use the quadratic formula to solve the equation.
x = (-b ± √(b^2 - 4ac)) / 2a = (-5 ± √(5^2 - 4(1)(6))) / 2(1) = (-5 ± √(25 - 24)) / 2 = (-5 ± √1) / 2 = (-5 ± 1) / 2
Solving for x, we get two possible values: x = (-5 + 1) / 2 = -2 and x = (-5 - 1) / 2 = -3.
Q: What if the quadratic equation has a complex root?
A: If the quadratic equation has a complex root, it means that the discriminant is negative. In this case, the quadratic formula will give you two complex roots.
Conclusion
In conclusion, quadratic equations and the discriminant are fundamental concepts in mathematics that have numerous real-world applications. By understanding the concept of the discriminant and how to find it, you can solve quadratic equations and determine the nature of their roots. We hope that this Q&A guide has been helpful in answering your questions about quadratic equations and the discriminant.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Discriminant" by Wolfram MathWorld
- [3] "Quadratic Formula" by Khan Academy
Glossary
- Quadratic Equation: A polynomial equation of degree two, which means the highest power of the variable is two.
- Discriminant: A value that can be calculated from the coefficients of a quadratic equation, which determines the nature of the roots of the equation.
- Quadratic Formula: A powerful tool for solving quadratic equations, which is given by x = (-b ± √(b^2 - 4ac)) / 2a.