(b) If $$ \alpha $$ and $$ \beta $$ are roots of the quadratic equation $$ x^{2}-2 x-15=0 $$. find (1) $$ \alpha+\beta $$ (II) $$ \alpha \beta $$ (III) $$ \frac{1}{\alpha}+\frac{1}{\beta} $$ (IV) $$ \frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} $$ (1) $$ \alpha-\beta $$

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We are given that $$ \alpha $$ and $$ \beta $$ are roots of the quadratic equation

$$
x^2 - 2x - 15 = 0.
$$

**Step 1. Identify the coefficients**

For the quadratic equation in the form

$$
ax^2 + bx + c = 0,
$$

we have: 
$$
a=1, \quad b=-2, \quad c=-15.
$$

**Step 2. Apply Viète’s formulas**

The sum and product of the roots are given by

$$
\alpha+\beta = -\frac{b}{a} \quad \text{and} \quad \alpha\beta = \frac{c}{a}.
$$

Thus,

$$
\alpha+\beta = -\frac{-2}{1} = 2,
$$

$$
\alpha\beta = \frac{-15}{1} = -15.
$$

**(i) $$ \alpha+\beta $$**

$$
\alpha+\beta = 2.
$$

**(ii) $$ \alpha \beta $$**

$$
\alpha \beta = -15.
$$

**(iii) $$ \frac{1}{\alpha}+\frac{1}{\beta} $$**

Using the common denominator we have

$$
\frac{1}{\alpha}+\frac{1}{\beta} = \frac{\beta+\alpha}{\alpha\beta}.
$$

Substitute the known values:

$$
\frac{1}{\alpha}+\frac{1}{\beta} = \frac{2}{-15} = -\frac{2}{15}.
$$

**(iv) $$ \frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} $$**

Write the expression as

$$
\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} = \frac{\beta^{2}+\alpha^{2}}{(\alpha\beta)^{2}}.
$$

First, compute $$ \alpha^2+\beta^2 $$ using the identity

$$
\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta.
$$

Substitute the known values:

$$
\alpha^2+\beta^2 = 2^2 - 2(-15) = 4 + 30 = 34.
$$

Then, since

$$
(\alpha\beta)^2 = (-15)^2 = 225,
$$

we have

$$
\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} = \frac{34}{225}.
$$

**(v) $$ \alpha - \beta $$**

We can find the difference between the roots using the formula:

$$
\alpha - \beta = \sqrt{(\alpha+\beta)^2 - 4\alpha\beta}.
$$

Substitute the values:

$$
\alpha - \beta = \sqrt{2^2 - 4(-15)} = \sqrt{4 + 60} = \sqrt{64} = 8.
$$

Thus, assuming $$\alpha$$ is the larger root, we have

$$
\alpha - \beta = 8.
$$
$$ \alpha + \beta = 2 $$, $$ \alpha \beta = -15 $$, $$ \frac{1}{\alpha} + \frac{1}{\beta} = -\frac{2}{15} $$, $$ \frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{34}{225} $$, and $$ \alpha - \beta = 8 $$.

(b) If and are roots of the quadratic equation . find
(1)
(II)
(III)
(IV)
(1)

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(b) If and are roots of the quadratic equation . find (1) (II) (III) (IV) (1)