If One Of The Roots Of Ax² + Bx + C Is Reciprocal Of Other Then-i) A=0ii) A=biii) A=civ) B=c

Mar 9, 2025 by ADMIN 94 views

If One of the Roots of ax² + bx + c is Reciprocal of the Other

Understanding the Problem

When dealing with quadratic equations of the form ax² + bx + c, we often encounter various properties and relationships between the coefficients and roots. One such interesting scenario is when one of the roots is the reciprocal of the other. In this article, we will delve into the implications of this condition and explore the relationships between the coefficients a, b, and c.

Reciprocal Roots: A Key Concept

To begin with, let's consider a quadratic equation with roots α and β. If one of the roots is the reciprocal of the other, we can express this relationship as α = 1/β or β = 1/α. This condition is crucial in understanding the properties of the quadratic equation and its coefficients.

Case i: a = 0

Let's start by examining the case where a = 0. In this scenario, the quadratic equation reduces to bx + c = 0. Since a = 0, the equation has a single root, which can be found by solving the linear equation bx + c = 0. However, this case does not satisfy the condition of having reciprocal roots, as there is only one root.

Case ii: a = b

Now, let's consider the case where a = b. In this scenario, the quadratic equation can be rewritten as ax² + ax + c = 0. By factoring out the common term a, we get a(x² + x + c/a) = 0. This equation has a root at x = -1, which is the reciprocal of the other root. Therefore, this case satisfies the condition of having reciprocal roots.

Case iii: a = c

Next, let's examine the case where a = c. In this scenario, the quadratic equation can be rewritten as ax² + bx + a = 0. By factoring out the common term a, we get a(x² + b/a x + 1) = 0. This equation has a root at x = -b/a, which is not necessarily the reciprocal of the other root. Therefore, this case does not satisfy the condition of having reciprocal roots.

Case iv: b = c

Finally, let's consider the case where b = c. In this scenario, the quadratic equation can be rewritten as ax² + bx + b = 0. By factoring out the common term b, we get b(ax²/a + x + 1) = 0. This equation has a root at x = -1, which is the reciprocal of the other root. Therefore, this case satisfies the condition of having reciprocal roots.

Conclusion

In conclusion, the condition of having reciprocal roots in a quadratic equation ax² + bx + c is satisfied when a = b or b = c. These cases result in the equation having roots that are reciprocals of each other. On the other hand, the cases a = 0 and a = c do not satisfy this condition. Understanding these relationships is crucial in solving quadratic equations and analyzing their properties.

Key Takeaways

The condition of having reciprocal roots in a quadratic equation ax² + bx + c is satisfied when a = b or b = c.
The cases a = 0 and a = c do not satisfy this condition.
Understanding the relationships between the coefficients and roots is crucial in solving quadratic equations and analyzing their properties.

Further Reading

For more information on quadratic equations and their properties, we recommend exploring the following topics:

Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution for the roots of a quadratic equation.
Discriminant: The discriminant is a key concept in quadratic equations, which determines the nature of the roots. It can be used to determine whether the roots are real or complex.
Quadratic Inequalities: Quadratic inequalities are an extension of quadratic equations, where the inequality sign is used instead of the equality sign. They can be used to solve problems involving quadratic expressions.

References

[1]: "Quadratic Equations" by Math Open Reference. A comprehensive online resource for quadratic equations and their properties.
[2]: "Quadratic Formula" by Khan Academy. A video tutorial on the quadratic formula and its application.
[3]: "Discriminant" by Wolfram MathWorld. A detailed article on the discriminant and its properties.

Glossary

Quadratic Equation: A polynomial equation of degree two, which can be written in the form ax² + bx + c = 0.
Roots: The solutions to a quadratic equation, which can be real or complex numbers.
Coefficients: The numerical constants in a quadratic equation, which can be a, b, or c.
Reciprocal Roots: Roots that are reciprocals of each other, which can be expressed as α = 1/β or β = 1/α.
Quadratic Equations: A Q&A Guide

Understanding Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their properties.

Q: What is a Quadratic Equation?

A: A quadratic equation is a polynomial equation of degree two, which can be written in the form ax² + bx + c = 0, where a, b, and c are numerical constants, and x is the variable.

Q: What are the Coefficients in a Quadratic Equation?

A: The coefficients in a quadratic equation are the numerical constants a, b, and c. The coefficient a is the coefficient of the squared term, the coefficient b is the coefficient of the linear term, and the coefficient c is the constant term.

Q: What are the Roots of a Quadratic Equation?

A: The roots of a quadratic equation are the solutions to the equation, which can be real or complex numbers. The roots can be found using the quadratic formula, which is x = (-b ± √(b² - 4ac)) / 2a.

Q: What is the Discriminant in a Quadratic Equation?

A: The discriminant in a quadratic equation is the expression b² - 4ac, which determines the nature of the roots. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: How Do I Find the Roots of a Quadratic Equation?

A: To find the roots of a quadratic equation, you can use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / 2a. You can also use factoring, completing the square, or the quadratic formula to find the roots.

Q: What is the Reciprocal of a Root?

A: The reciprocal of a root is a root that is the inverse of the original root. For example, if the root is 2, the reciprocal is 1/2.

Q: How Do I Determine If One of the Roots is the Reciprocal of the Other?

A: To determine if one of the roots is the reciprocal of the other, you can use the following conditions:

If a = b, then one of the roots is the reciprocal of the other.
If b = c, then one of the roots is the reciprocal of the other.

Q: What is the Significance of Reciprocal Roots in Quadratic Equations?

A: Reciprocal roots in quadratic equations have significant implications in various fields, including physics, engineering, and economics. They can be used to model real-world problems, such as population growth, chemical reactions, and electrical circuits.

Q: Can You Provide Examples of Quadratic Equations with Reciprocal Roots?

A: Yes, here are some examples of quadratic equations with reciprocal roots:

x² + x + 1 = 0 (a = b)
x² + 2x + 1 = 0 (b = c)

Q: How Do I Use Quadratic Equations in Real-World Problems?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. They can be used to model real-world problems, such as:

Population growth: Quadratic equations can be used to model population growth, where the population is represented by the variable x.
Chemical reactions: Quadratic equations can be used to model chemical reactions, where the concentration of a substance is represented by the variable x.
Electrical circuits: Quadratic equations can be used to model electrical circuits, where the voltage and current are represented by the variables x and y.