If the zeroes of the polynomial `x^2+px+q` are double the value to the zeroes `2x^2-5x-3` find the value of p and q

Answer

To find the values of \( p \) and \( q \) for the polynomial \( x^2 + px + q \), given that its zeroes are double the zeroes of the polynomial \( 2x^2 - 5x - 3 \), we can follow these steps:

### Step 1: Identify the zeroes of the polynomial \( 2x^2 - 5x - 3 \)

To find the zeroes (roots) of the polynomial \( 2x^2 - 5x - 3 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 2 \), \( b = -5 \), and \( c = -3 \).

### Step 2: Calculate the discriminant

First, we calculate the discriminant \( D \):

\[ D = b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49 \]

### Step 3: Find the roots

Now, substituting the values into the quadratic formula:

\[ x = \frac{5 \pm \sqrt{49}}{2 \cdot 2} = \frac{5 \pm 7}{4} \]

Calculating the two possible values:

1. \( x_1 = \frac{12}{4} = 3 \)
2. \( x_2 = \frac{-2}{4} = -\frac{1}{2} \)

Thus, the roots of the polynomial \( 2x^2 - 5x - 3 \) are \( \alpha = 3 \) and \( \beta = -\frac{1}{2} \).

### Step 4: Determine the zeroes of the polynomial \( x^2 + px + q \)

Since the zeroes of \( x^2 + px + q \) are double the zeroes of \( 2x^2 - 5x - 3 \), we have:

\[ 2\alpha = 2 \cdot 3 = 6 \quad \text{and} \quad 2\beta = 2 \cdot \left(-\frac{1}{2}\right) = -1 \]

Thus, the zeroes of \( x^2 + px + q \) are \( 6 \) and \( -1 \).

### Step 5: Find the sum and product of the roots

Using the properties of polynomials, we know:

- The sum of the roots \( (6 + (-1)) = 5 \)
- The product of the roots \( (6 \cdot (-1)) = -6 \)

### Step 6: Relate the sum and product to \( p \) and \( q \)

For the polynomial \( x^2 + px + q \):

- The sum of the roots is given by \( -p \):
\[ -p = 5 \implies p = -5 \]

- The product of the roots is given by \( q \):
\[ q = -6 \]

### Conclusion

Thus, the values of \( p \) and \( q \) are:

\[ p = -5, \quad q = -6 \]