For each of the following find a quadratic polynomial whose sum and product respectively of the zeroes are as given .Also find the zeroes of these polynomials by factorisation.
`(-3)/(2sqrt(5)),(-1)/(2)`

Answer

To find a quadratic polynomial whose sum and product of the zeroes are given, we can use the following steps:

### Step 1: Identify the sum and product of the zeroes
The sum of the zeroes \( \alpha + \beta \) is given as:
\[ \alpha + \beta = -\frac{3}{2\sqrt{5}} \]
The product of the zeroes \( \alpha \beta \) is given as:
\[ \alpha \beta = -\frac{1}{2} \]

### Step 2: Write the general form of the quadratic polynomial
The general form of a quadratic polynomial based on the sum and product of its roots is:
\[ p(x) = x^2 - (\text{sum of zeroes}) \cdot x + (\text{product of zeroes}) \]
Substituting the values we have:
\[ p(x) = x^2 - \left(-\frac{3}{2\sqrt{5}}\right) x + \left(-\frac{1}{2}\right) \]

### Step 3: Simplify the polynomial
This simplifies to:
\[ p(x) = x^2 + \frac{3}{2\sqrt{5}} x - \frac{1}{2} \]

### Step 4: Eliminate the fraction by multiplying through by \( 2\sqrt{5} \)
To eliminate the fractions, we can multiply the entire polynomial by \( 2\sqrt{5} \):
\[ p(x) = 2\sqrt{5} \cdot x^2 + 3x - \sqrt{5} \]

### Step 5: Factor the polynomial
Now we need to factor the polynomial \( 2\sqrt{5} x^2 + 3x - \sqrt{5} \). We can use the method of splitting the middle term.

1. We need two numbers that multiply to \( 2\sqrt{5} \cdot (-\sqrt{5}) = -10 \) and add to \( 3 \).
2. The numbers \( 5 \) and \( -2 \) work since \( 5 \cdot (-2) = -10 \) and \( 5 + (-2) = 3 \).

Now we can rewrite the middle term:
\[ p(x) = 2\sqrt{5} x^2 + 5x - 2x - \sqrt{5} \]

### Step 6: Group the terms
Grouping the terms gives us:
\[ = (2\sqrt{5} x^2 + 5x) + (-2x - \sqrt{5}) \]
Factoring out common terms:
\[ = x(2\sqrt{5} x + 5) - \sqrt{5}(2x + 1) \]

### Step 7: Factor by grouping
Now we can factor by grouping:
\[ = (2\sqrt{5} x + 5)(x - \frac{\sqrt{5}}{2}) \]

### Step 8: Find the zeroes
To find the zeroes, we set each factor to zero:
1. \( 2\sqrt{5} x + 5 = 0 \)
\[ 2\sqrt{5} x = -5 \implies x = -\frac{5}{2\sqrt{5}} = -\frac{\sqrt{5}}{2} \]

2. \( x - \frac{\sqrt{5}}{2} = 0 \)
\[ x = \frac{\sqrt{5}}{2} \]

### Final Result
The zeroes of the polynomial are:
\[ x = -\frac{\sqrt{5}}{2}, \quad x = \frac{\sqrt{5}}{2} \]