If $\alpha$ And $\beta$ Are The Roots Of The Equation $4 + 3x - 2x^2 = 0$, Find The Value Of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$.

Introduction

In this article, we will explore the problem of finding the value of the sum of the reciprocals of the squares of the roots of a given quadratic equation. The equation in question is $4 + 3x - 2x^2 = 0$, and we are asked to find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$, where $\alpha$ and $\beta$ are the roots of the equation.

The Quadratic Formula

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

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our case, the equation is $4 + 3x - 2x^2 = 0$, which can be rewritten as $-2x^2 + 3x + 4 = 0$. Comparing this to the general form, we have $a = -2$, $b = 3$, and $c = 4$.

Finding the Roots

Using the quadratic formula, we can find the roots of the equation:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(-2)(4)}}{2(-2)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-3 \pm \sqrt{9 + 32}}{-4}$$

$$x = \frac{-3 \pm \sqrt{41}}{-4}$$

Therefore, the roots of the equation are:

$$\alpha = \frac{-3 + \sqrt{41}}{-4}$$

$$\beta = \frac{-3 - \sqrt{41}}{-4}$$

Finding the Sum of Reciprocals of Squares

We are asked to find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$. To do this, we can start by finding the values of $\alpha^2$ and $\beta^2$.

Using the fact that $\alpha = \frac{-3 + \sqrt{41}}{-4}$ and $\beta = \frac{-3 - \sqrt{41}}{-4}$, we can square both expressions to get:

$$\alpha^2 = \left(\frac{-3 + \sqrt{41}}{-4}\right)^2$$

$$\beta^2 = \left(\frac{-3 - \sqrt{41}}{-4}\right)^2$$

Simplifying the expressions, we get:

$$\alpha^2 = \frac{9 - 6\sqrt{41} + 41}{16}$$

$$\beta^2 = \frac{9 + 6\sqrt{41} + 41}{16}$$

Therefore, the values of $\alpha^2$ and $\beta^2$ are:

$$\alpha^2 = \frac{50 - 6\sqrt{41}}{16}$$

$$\beta^2 = \frac{50 + 6\sqrt{41}}{16}$$

Finding the Sum of Reciprocals

Now that we have the values of $\alpha^2$ and $\beta^2$, we can find the sum of their reciprocals:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{1}{\frac{50 - 6\sqrt{41}}{16}} + \frac{1}{\frac{50 + 6\sqrt{41}}{16}}$$

Simplifying the expression, we get:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{16}{50 - 6\sqrt{41}} + \frac{16}{50 + 6\sqrt{41}}$$

Using the fact that $\frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab}$, we can simplify the expression further:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{(50 - 6\sqrt{41}) + (50 + 6\sqrt{41})}{(50 - 6\sqrt{41})(50 + 6\sqrt{41})}$$

Simplifying the numerator and denominator, we get:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{100}{2500 - 216}$$

Simplifying the expression further, we get:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{100}{2284}$$

Therefore, the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$ is $\frac{100}{2284}$.

Conclusion

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Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for finding the roots of a quadratic equation. It states that for an equation of the form $ax^2 + bx + c = 0$, the roots are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to find the roots of a quadratic equation?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, you can plug these values into the formula to find the roots.

Q: What is the difference between $\alpha$ and $\beta$ in the context of the quadratic equation?

A: In the context of the quadratic equation, $\alpha$ and $\beta$ are the roots of the equation. They are the values of $x$ that satisfy the equation.

Q: How do I find the sum of the reciprocals of the squares of the roots of a quadratic equation?

A: To find the sum of the reciprocals of the squares of the roots of a quadratic equation, you need to first find the values of $\alpha^2$ and $\beta^2$. Then, you can find the sum of their reciprocals using the formula:

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{1}{\frac{50 - 6\sqrt{41}}{16}} + \frac{1}{\frac{50 + 6\sqrt{41}}{16}}$$

Q: What is the final answer to the problem of finding the sum of the reciprocals of the squares of the roots of a quadratic equation?

A: The final answer to the problem of finding the sum of the reciprocals of the squares of the roots of a quadratic equation is $\frac{100}{2284}$.

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

A: Yes, you can use the quadratic formula to find the roots of any quadratic equation. However, you need to make sure that the equation is in the form $ax^2 + bx + c = 0$.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not plugging the values of $a$, $b$, and $c$ into the formula correctly
• Not simplifying the expression under the square root correctly
• Not simplifying the final expression correctly

Q: How can I practice using the quadratic formula to find the roots of quadratic equations?

A: You can practice using the quadratic formula to find the roots of quadratic equations by working through examples and exercises. You can also use online resources and calculators to help you practice.

Q: What are some real-world applications of the quadratic formula?

A: The quadratic formula has many real-world applications, including:

• Finding the maximum or minimum value of a quadratic function
• Determining the stability of a system
• Modeling population growth or decline
• Solving problems in physics, engineering, and economics

Q: Can I use the quadratic formula to solve systems of linear equations?

A: No, you cannot use the quadratic formula to solve systems of linear equations. The quadratic formula is used to solve quadratic equations, not systems of linear equations.