Quadratic Equation and its Roots

Oct 13, 2025 by windailys.com 48 views

Introduction

In algebra, the sum and product of roots of a quadratic equation are essential concepts. Given a quadratic equation in the form of ax² + bx + c = 0, the sum of the roots (a and b) is given by -b/a, and the product of the roots is given by c/a. In this article, we will explore the concept of sum and product of roots and use it to evaluate the expression 1 /2a+ẞ + 1 /a+2ẞ.

Quadratic Equation and its Roots

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (x) is two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation.

In the given equation 3x² + 14x + 7 = 0, the coefficients are a = 3, b = 14, and c = 7. To find the roots of the equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.

Sum and Product of Roots

The sum of the roots (a and b) of a quadratic equation is given by -b/a. In this case, the sum of the roots is -14/3.

The product of the roots (a and b) of a quadratic equation is given by c/a. In this case, the product of the roots is 7/3.

Evaluating the Expression

Now, let's evaluate the expression 1 /2a+ẞ + 1 /a+2ẞ. To do this, we need to find the values of 2a+ẞ and a+2ẞ.

Finding 2a+ẞ

To find 2a+ẞ, we can use the fact that the sum of the roots is -b/a. In this case, the sum of the roots is -14/3. We can write:

2a+ẞ = -b/a 2a+ẞ = -14/3

Finding a+2ẞ

To find a+2ẞ, we can use the fact that the product of the roots is c/a. In this case, the product of the roots is 7/3. We can write:

a+2ẞ = c/a a+2ẞ = 7/3

Evaluating the Expression

Now that we have found the values of 2a+ẞ and a+2ẞ, we can evaluate the expression 1 /2a+ẞ + 1 /a+2ẞ.

1 /2a+ẞ + 1 /a+2ẞ = 1 /(-14/3) + 1 / (7/3) = -3/14 + 3/7 = -3/14 + 6/14 = 3/14

Conclusion

In this article, we have evaluated the expression 1 /2a+ẞ + 1 /a+2ẞ given that a and b are roots of the equation 3x² + 14x + 7 = 0. We have used the concepts of sum and product of roots to find the values of 2a+ẞ and a+2ẞ, and then evaluated the expression. The final answer is 3/14.

References

[1] Khan Academy. (n.d.). Quadratic equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7f0d-quadratic-equations
[2] Math Open Reference. (n.d.). Quadratic equation. Retrieved from https://www.mathopenref.com/quadratic.html

Introduction

In our previous article, we evaluated the expression 1 /2a+ẞ + 1 /a+2ẞ given that a and b are roots of the equation 3x² + 14x + 7 = 0. We used the concepts of sum and product of roots to find the values of 2a+ẞ and a+2ẞ, and then evaluated the expression. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is the sum of the roots of a quadratic equation?

A: The sum of the roots of a quadratic equation is given by -b/a.

Q: What is the product of the roots of a quadratic equation?

A: The product of the roots of a quadratic equation is given by c/a.

Q: How do I find the values of 2a+ẞ and a+2ẞ?

A: To find the values of 2a+ẞ and a+2ẞ, you can use the fact that the sum of the roots is -b/a and the product of the roots is c/a.

Q: What is the value of 1 /2a+ẞ + 1 /a+2ẞ?

A: The value of 1 /2a+ẞ + 1 /a+2ẞ is 3/14.

Q: Can I use the quadratic formula to find the roots of the equation?

A: Yes, you can use the quadratic formula to find the roots of the equation. The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.

Q: What is the significance of the sum and product of roots in a quadratic equation?

A: The sum and product of roots are essential concepts in algebra. They are used to find the values of 2a+ẞ and a+2ẞ, and then evaluate expressions like 1 /2a+ẞ + 1 /a+2ẞ.

Q: Can I apply the concepts of sum and product of roots to other types of equations?

A: Yes, the concepts of sum and product of roots can be applied to other types of equations, not just quadratic equations.

Conclusion

In this article, we have answered some frequently asked questions related to the topic of evaluating the expression 1 /2a+ẞ + 1 /a+2ẞ given that a and b are roots of the equation 3x² + 14x + 7 = 0. We have used the concepts of sum and product of roots to find the values of 2a+ẞ and a+2ẞ, and then evaluated the expression.

References

[1] Khan Academy. (n.d.). Quadratic equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7f0d-quadratic-equations
[2] Math Open Reference. (n.d.). Quadratic equation. Retrieved from https://www.mathopenref.com/quadratic.html

Additional Resources

[1] Wolfram Alpha. (n.d.). Quadratic equation. Retrieved from https://www.wolframalpha.com/input/?i=quadratic+equation
[2] Mathway. (n.d.). Quadratic equation. Retrieved from https://www.mathway.com/subjects/quadratic-equation

We hope this article has been helpful in answering your questions related to the topic of evaluating the expression 1 /2a+ẞ + 1 /a+2ẞ given that a and b are roots of the equation 3x² + 14x + 7 = 0. If you have any further questions or need additional help, please don't hesitate to ask.