Construct a quadratic equation whose roots are (1) $$ \begin{array}{l} ext { (2) } \alpha+\frac{1}{\beta}, \beta+\frac{1}{\alpha}\end{array} $$ (3) $$ \alpha^{3} $$ and $$ \beta^{3} $$ Assignment

Question

Construct a quadratic equation whose roots are (1) $$ \begin{array}{l}	ext { (2) } \alpha+\frac{1}{\beta}, \beta+\frac{1}{\alpha}\end{array} $$ (3) $$ \alpha^{3} $$ and $$ \beta^{3} $$ Assignment

UpStudy AI · Accepted Answer

To construct a quadratic equation whose roots are given as $$ r_1 = \alpha + \frac{1}{\beta} $$, $$ r_2 = \beta + \frac{1}{\alpha} $$, we can use Vieta's formulas. According to Vieta's, for a quadratic equation of the form $$ x^2 - (r_1 + r_2)x + r_1r_2 = 0 $$, the sum of the roots $$ r_1 + r_2 $$ and the product of the roots $$ r_1r_2 $$ can be used to form the equation.

### Step 1: Calculate the sum of the roots $$ r_1 + r_2 $$

$$
r_1 + r_2 = \left( \alpha + \frac{1}{\beta} \right) + \left( \beta + \frac{1}{\alpha} \right) = \alpha + \beta + \frac{1}{\beta} + \frac{1}{\alpha}
$$

### Step 2: Calculate the product of the roots $$ r_1r_2 $$

$$
r_1r_2 = \left( \alpha + \frac{1}{\beta} \right) \left( \beta + \frac{1}{\alpha} \right)
$$

Expanding this product:

$$
r_1r_2 = \alpha\beta + \frac{\alpha}{\alpha} + \frac{\beta}{\beta} + \frac{1}{\alpha\beta} = \alpha\beta + 1 + 1 + \frac{1}{\alpha\beta} = \alpha\beta + 2 + \frac{1}{\alpha\beta}
$$

### Step 3: Form the quadratic equation

Now we can substitute the sum and product into the quadratic equation:

$$
x^2 - (r_1 + r_2)x + r_1r_2 = 0
$$

Substituting the values we calculated:

$$
x^2 - \left( \alpha + \beta + \frac{1}{\beta} + \frac{1}{\alpha} \right)x + \left( \alpha\beta + 2 + \frac{1}{\alpha\beta} \right) = 0
$$

### Final Quadratic Equation

Thus, the quadratic equation whose roots are $$ \alpha + \frac{1}{\beta} $$ and $$ \beta + \frac{1}{\alpha} $$ is:

$$
x^2 - \left( \alpha + \beta + \frac{1}{\beta} + \frac{1}{\alpha} \right)x + \left( \alpha\beta + 2 + \frac{1}{\alpha\beta} \right) = 0
$$

This is the required quadratic equation.
The quadratic equation with roots $$ \alpha + \frac{1}{\beta} $$ and $$ \beta + \frac{1}{\alpha} $$ is: $$ x^2 - \left( \alpha + \beta + \frac{1}{\beta} + \frac{1}{\alpha} \right)x + \left( \alpha\beta + 2 + \frac{1}{\alpha\beta} \right) = 0 $$

Construct a quadratic equation whose roots are (1)
(3) and Assignment

Upstudy AI Solution

Answer

Solution

Step 1: Calculate the sum of the roots

Step 2: Calculate the product of the roots

Step 3: Form the quadratic equation

Final Quadratic Equation

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Construct a quadratic equation whose roots are (1) (3) and Assignment