Form the quadratic equations whose roots are given below
i) `7+-2sqrt5`
ii) `(a)/(b), (-b)/(a)` (`a!=0, b != 0`)
iii) `(p-q)/(p+q), -((p+q)/(p-q))` (`p != +-q`)
iv) `-3 +- 5i`
v) 2, 5 vi) `2+sqrt3, 2-sqrt3`
vii) -a+ib, -a-ib

Answer

To form the quadratic equations from the given roots, we will use the fact that if \( \alpha \) and \( \beta \) are the roots of a quadratic equation, then the equation can be expressed as:

\[ x^2 - sx + p = 0 \]

where \( s = \alpha + \beta \) (sum of the roots) and \( p = \alpha \cdot \beta \) (product of the roots).

Now, let's solve each part step by step.

### Part i: Roots \( 7 \pm 2\sqrt{5} \)

1. **Identify the roots**:
- \( \alpha = 7 + 2\sqrt{5} \)
- \( \beta = 7 - 2\sqrt{5} \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = (7 + 2\sqrt{5}) + (7 - 2\sqrt{5}) = 14 \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = (7 + 2\sqrt{5})(7 - 2\sqrt{5}) = 7^2 - (2\sqrt{5})^2 = 49 - 20 = 29 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 - 14x + 29 = 0 \]

### Part ii: Roots \( \frac{a}{b}, -\frac{b}{a} \)

1. **Identify the roots**:
- \( \alpha = \frac{a}{b} \)
- \( \beta = -\frac{b}{a} \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = \frac{a}{b} - \frac{b}{a} = \frac{a^2 - b^2}{ab} \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = \frac{a}{b} \cdot \left(-\frac{b}{a}\right) = -1 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 - \left(\frac{a^2 - b^2}{ab}\right)x - 1 = 0 \]
Multiplying through by \( ab \):
\[ abx^2 - (a^2 - b^2)x - ab = 0 \]

### Part iii: Roots \( \frac{p-q}{p+q}, -\frac{p+q}{p-q} \)

1. **Identify the roots**:
- \( \alpha = \frac{p-q}{p+q} \)
- \( \beta = -\frac{p+q}{p-q} \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = \frac{p-q}{p+q} - \frac{p+q}{p-q} \]
Finding a common denominator:
\[ s = \frac{(p-q)(p-q) - (p+q)(p+q)}{(p+q)(p-q)} = \frac{(p^2 - 2pq + q^2) - (p^2 + 2pq + q^2)}{p^2 - q^2} = \frac{-4pq}{p^2 - q^2} \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = \frac{p-q}{p+q} \cdot \left(-\frac{p+q}{p-q}\right) = -1 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 + \frac{4pq}{p^2 - q^2}x + 1 = 0 \]
Multiplying through by \( p^2 - q^2 \):
\[ (p^2 - q^2)x^2 + 4pqx + (p^2 - q^2) = 0 \]

### Part iv: Roots \( -3 \pm 5i \)

1. **Identify the roots**:
- \( \alpha = -3 + 5i \)
- \( \beta = -3 - 5i \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = (-3 + 5i) + (-3 - 5i) = -6 \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = (-3 + 5i)(-3 - 5i) = (-3)^2 + (5)^2 = 9 + 25 = 34 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 + 6x + 34 = 0 \]

### Part v: Roots \( 2, 5 \)

1. **Identify the roots**:
- \( \alpha = 2 \)
- \( \beta = 5 \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = 2 + 5 = 7 \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = 2 \cdot 5 = 10 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 - 7x + 10 = 0 \]

### Part vi: Roots \( 2+\sqrt{3}, 2-\sqrt{3} \)

1. **Identify the roots**:
- \( \alpha = 2 + \sqrt{3} \)
- \( \beta = 2 - \sqrt{3} \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = (2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 - 4x + 1 = 0 \]

### Part vii: Roots \( -a + ib, -a - ib \)

1. **Identify the roots**:
- \( \alpha = -a + ib \)
- \( \beta = -a - ib \)

2. **Calculate the sum \( s \)**:
\[ s = \alpha + \beta = (-a + ib) + (-a - ib) = -2a \]

3. **Calculate the product \( p \)**:
\[ p = \alpha \cdot \beta = (-a + ib)(-a - ib) = a^2 + b^2 \]

4. **Form the quadratic equation**:
\[ x^2 - sx + p = 0 \implies x^2 + 2ax + (a^2 + b^2) = 0 \]

### Summary of Quadratic Equations:
1. \( x^2 - 14x + 29 = 0 \)
2. \( abx^2 - (a^2 - b^2)x - ab = 0 \)
3. \( (p^2 - q^2)x^2 + 4pqx + (p^2 - q^2) = 0 \)
4. \( x^2 + 6x + 34 = 0 \)
5. \( x^2 - 7x + 10 = 0 \)
6. \( x^2 - 4x + 1 = 0 \)
7. \( x^2 + 2ax + (a^2 + b^2) = 0 \)