If Α \alpha Α And Β \beta Β Are The Zeroes Of 2 X 2 + X − 1 2x^2 + X - 1 2 X 2 + X − 1 , Then Find 1 Α + 1 Β \frac{1}{\alpha} + \frac{1}{\beta} Α 1 ​ + Β 1 ​ .

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Introduction

In algebra, 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 ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and aa cannot be zero. In this article, we will focus on solving a quadratic equation and finding the sum of the reciprocals of its roots.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will be used to find the roots of the given quadratic equation.

The Given Quadratic Equation

The given quadratic equation is 2x2+x−1=02x^2 + x - 1 = 0. We are asked to find the sum of the reciprocals of its roots, which means we need to find 1α+1β\frac{1}{\alpha} + \frac{1}{\beta}.

Finding the Roots of the Quadratic Equation

To find the roots of the quadratic equation, we can use the quadratic formula. Plugging in the values of aa, bb, and cc into the formula, we get:

x=−1±1+84x = \frac{-1 \pm \sqrt{1 + 8}}{4}

Simplifying the expression under the square root, we get:

x=−1±94x = \frac{-1 \pm \sqrt{9}}{4}

This simplifies to:

x=−1±34x = \frac{-1 \pm 3}{4}

Therefore, the roots of the quadratic equation are:

α=−1+34=12\alpha = \frac{-1 + 3}{4} = \frac{1}{2}

β=−1−34=−1\beta = \frac{-1 - 3}{4} = -1

Finding the Sum of the Reciprocals of the Roots

Now that we have found the roots of the quadratic equation, we can find the sum of the reciprocals of the roots. The sum of the reciprocals of the roots is given by:

1α+1β=112+1−1\frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{\frac{1}{2}} + \frac{1}{-1}

Simplifying the expression, we get:

1α+1β=2−1\frac{1}{\alpha} + \frac{1}{\beta} = 2 - 1

Therefore, the sum of the reciprocals of the roots is:

1α+1β=1\frac{1}{\alpha} + \frac{1}{\beta} = 1

Conclusion

In this article, we solved a quadratic equation and found the sum of the reciprocals of its roots. We used the quadratic formula to find the roots of the equation and then found the sum of the reciprocals of the roots. The sum of the reciprocals of the roots is a useful concept in algebra and is used in many mathematical applications.

Final Answer

The final answer is 1\boxed{1}.

Additional Resources

For more information on solving quadratic equations and finding the sum of the reciprocals of its roots, please refer to the following resources:

Introduction

Quadratic equations are a fundamental concept in algebra, and understanding them is crucial for solving a wide range of mathematical problems. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

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. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and aa cannot be zero.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods to solve a quadratic equation, including:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the Quadratic Formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula. Then, simplify the expression under the square root and solve for xx.

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: $\alpha + \beta = -\frac{b}{a}$

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: $\alpha \beta = \frac{c}{a}$

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

A: To find the sum of the reciprocals of the roots of a quadratic equation, you need to find the sum of the reciprocals of the roots, which is given by: $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$

Q: What is the Difference of the Roots of a Quadratic Equation?

A: The difference of the roots of a quadratic equation is given by: $\alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta}$

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic equations and how to solve them. We covered topics such as the quadratic formula, sum and product of roots, and difference of roots. We hope this guide has been helpful in your understanding of quadratic equations.

Additional Resources

For more information on quadratic equations, please refer to the following resources:

Note: The Q&A format is a common way to present information in a clear and concise manner. The additional resources section provides links to external resources that can be used for further learning.