9. If $$ \alpha $$ and $$ \beta $$ are the roots of $$ a x^{2}+b x+c=0 $$, find the equations whose roors $$ \begin{array}{lll}\alpha \beta^{-1} \cdot \beta \alpha^{-1} & \alpha^{2} \cdot \beta^{2} & ext { c. } \alpha^{2}+\beta^{-2}+\beta^{-2} \ \alpha^{3} \cdot \beta^{3} & \sqrt{\frac{\alpha}{\beta}} \cdot \sqrt{\frac{\beta}{\alpha}} & ext { f. }(a \alpha+b)^{-1},(a \beta \pm b)^{-1}\end{array} $$

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9. If $$ \alpha $$ and $$ \beta $$ are the roots of $$ a x^{2}+b x+c=0 $$, find the equations whose roors $$ \begin{array}{lll}\alpha \beta^{-1} \cdot \beta \alpha^{-1} & \alpha^{2} \cdot \beta^{2} & 	ext { c. } \alpha^{2}+\beta^{-2}+\beta^{-2} \ \alpha^{3} \cdot \beta^{3} & \sqrt{\frac{\alpha}{\beta}} \cdot \sqrt{\frac{\beta}{\alpha}} & 	ext { f. }(a \alpha+b)^{-1},(a \beta \pm b)^{-1}\end{array} $$

UpStudy AI · Accepted Answer

To find the equations whose roots are given expressions involving the roots $$ \alpha $$ and $$ \beta $$ of the quadratic equation $$ ax^2 + bx + c = 0 $$, we will analyze each expression one by one.

1. **Roots: $$ \alpha \beta^{-1} \cdot \beta \alpha^{-1} $$**

This expression simplifies to:
   $$
   \alpha \beta^{-1} \cdot \beta \alpha^{-1} = 1
   $$
   The equation whose root is $$ 1 $$ is:
   $$
   x - 1 = 0
   $$

2. **Roots: $$ \alpha^2 \cdot \beta^2 $$**

This expression simplifies to:
   $$
   \alpha^2 \cdot \beta^2 = (\alpha \beta)^2
   $$
   Using Vieta's formulas, we know $$ \alpha \beta = \frac{c}{a} $$. Therefore:
   $$
   \alpha^2 \cdot \beta^2 = \left(\frac{c}{a}\right)^2
   $$
   The equation whose root is $$ \left(\frac{c}{a}\right)^2 $$ is:
   $$
   x - \left(\frac{c}{a}\right)^2 = 0
   $$

3. **Roots: $$ \alpha^2 + \beta^{-2} + \beta^{-2} $$**

This expression simplifies to:
   $$
   \alpha^2 + 2\beta^{-2} = \alpha^2 + 2\frac{1}{\beta^2}
   $$
   We can express $$ \beta^2 $$ in terms of $$ \alpha $$ and $$ \beta $$:
   $$
   \beta^2 = \frac{c^2}{a^2}
   $$
   Thus:
   $$
   \beta^{-2} = \frac{a^2}{c^2}
   $$
   Therefore:
   $$
   \alpha^2 + 2\beta^{-2} = \alpha^2 + 2\frac{a^2}{c^2}
   $$
   The equation whose root is $$ \alpha^2 + 2\frac{a^2}{c^2} $$ is:
   $$
   x - \left(\alpha^2 + 2\frac{a^2}{c^2}\right) = 0
   $$

4. **Roots: $$ \alpha^3 \cdot \beta^3 $$**

This expression simplifies to:
   $$
   \alpha^3 \cdot \beta^3 = (\alpha \beta)^3 = \left(\frac{c}{a}\right)^3
   $$
   The equation whose root is $$ \left(\frac{c}{a}\right)^3 $$ is:
   $$
   x - \left(\frac{c}{a}\right)^3 = 0
   $$

5. **Roots: $$ \sqrt{\frac{\alpha}{\beta}} \cdot \sqrt{\frac{\beta}{\alpha}} $$**

This expression simplifies to:
   $$
   \sqrt{\frac{\alpha}{\beta}} \cdot \sqrt{\frac{\beta}{\alpha}} = 1
   $$
   The equation whose root is $$ 1 $$ is:
   $$
   x - 1 = 0
   $$

6. **Roots: $$ (a \alpha + b)^{-1}, (a \beta \pm b)^{-1} $$**

The roots can be expressed as:
   $$
   \frac{1}{a \alpha + b} \quad \text{and} \quad \frac{1}{a \beta + b}
   $$
   The equations whose roots are $$ \frac{1}{a \alpha + b} $$ and $$ \frac{1}{a \beta + b} $$ can be derived from the original quadratic equation. The roots can be expressed as:
   $$
   x(a \alpha + b) = 1 \quad \text{and} \quad x(a \beta + b) = 1
   $$
   Thus, the equations are:
   $$
   ax + b = \frac{1}{\alpha} \quad \text{and} \quad ax + b = \frac{1}{\beta}
   $$

In summary, the equations corresponding to the roots are:

1. $$ x - 1 = 0 $$
2. $$ x - \left(\frac{c}{a}\right)^2 = 0 $$
3. $$ x - \left(\alpha^2 + 2\frac{a^2}{c^2}\right) = 0 $$
4. $$ x - \left(\frac{c}{a}\right)^3 = 0 $$
5. $$ x - 1 = 0 $$
6. $$ ax + b - \frac{1}{\alpha} = 0 $$ and $$ ax + b - \frac{1}{\beta} = 0 $$
1. $$ x - 1 = 0 $$ 2. $$ x - \left(\frac{c}{a}\right)^2 = 0 $$ 3. $$ x - \left(\alpha^2 + 2\frac{a^2}{c^2}\right) = 0 $$ 4. $$ x - \left(\frac{c}{a}\right)^3 = 0 $$ 5. $$ x - 1 = 0 $$ 6. $$ ax + b - \frac{1}{\alpha} = 0 $$ and $$ ax + b - \frac{1}{\beta} = 0 $$

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