The roots of the quadratic equation `4x^(2)-(5a+1)x+5a=0`, are p and q. if q=1+p, calculate the possible values of a,p and q.

Answer

To solve the problem step by step, we start with the given quadratic equation and the relationship between the roots.

### Given:
The quadratic equation is:
\[ 4x^2 - (5a + 1)x + 5a = 0 \]

The roots of this equation are \( p \) and \( q \), with the relationship:
\[ q = 1 + p \]

### Step 1: Identify coefficients
From the quadratic equation, we identify:
- \( a = 4 \)
- \( b = -(5a + 1) \)
- \( c = 5a \)

### Step 2: Use the sum and product of roots
The sum of the roots \( p + q \) can be calculated using the formula:
\[ p + q = -\frac{b}{a} = -\frac{-(5a + 1)}{4} = \frac{5a + 1}{4} \]

The product of the roots \( pq \) is given by:
\[ pq = \frac{c}{a} = \frac{5a}{4} \]

### Step 3: Substitute \( q \) in terms of \( p \)
Using the relationship \( q = 1 + p \), we substitute this into the sum of roots:
\[ p + (1 + p) = \frac{5a + 1}{4} \]
This simplifies to:
\[ 2p + 1 = \frac{5a + 1}{4} \]

### Step 4: Solve for \( p \)
Now, we can isolate \( p \):
\[ 2p = \frac{5a + 1}{4} - 1 \]
\[ 2p = \frac{5a + 1 - 4}{4} \]
\[ 2p = \frac{5a - 3}{4} \]
\[ p = \frac{5a - 3}{8} \]

### Step 5: Find \( q \)
Now substituting \( p \) back to find \( q \):
\[ q = 1 + p = 1 + \frac{5a - 3}{8} \]
\[ q = \frac{8 + 5a - 3}{8} = \frac{5a + 5}{8} \]

### Step 6: Use the product of roots
Now we use the product of the roots:
\[ pq = \frac{5a}{4} \]
Substituting the values of \( p \) and \( q \):
\[ \left(\frac{5a - 3}{8}\right) \left(\frac{5a + 5}{8}\right) = \frac{5a}{4} \]

### Step 7: Simplify and solve for \( a \)
Multiplying both sides by \( 64 \) to eliminate the denominators:
\[ (5a - 3)(5a + 5) = 80a \]
Expanding the left side:
\[ 25a^2 + 25a - 15a - 15 = 80a \]
This simplifies to:
\[ 25a^2 - 70a - 15 = 0 \]

### Step 8: Solve the quadratic equation for \( a \)
Using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( A = 25, B = -70, C = -15 \):
\[ a = \frac{70 \pm \sqrt{(-70)^2 - 4 \cdot 25 \cdot (-15)}}{2 \cdot 25} \]
\[ a = \frac{70 \pm \sqrt{4900 + 1500}}{50} \]
\[ a = \frac{70 \pm \sqrt{6400}}{50} \]
\[ a = \frac{70 \pm 80}{50} \]

Calculating the two possible values:
1. \( a = \frac{150}{50} = 3 \)
2. \( a = \frac{-10}{50} = -\frac{1}{5} \)

### Step 9: Calculate \( p \) and \( q \) for both values of \( a \)
1. For \( a = 3 \):
- \( p = \frac{5(3) - 3}{8} = \frac{15 - 3}{8} = \frac{12}{8} = \frac{3}{2} \)
- \( q = \frac{5(3) + 5}{8} = \frac{15 + 5}{8} = \frac{20}{8} = \frac{5}{2} \)

2. For \( a = -\frac{1}{5} \):
- \( p = \frac{5(-\frac{1}{5}) - 3}{8} = \frac{-1 - 3}{8} = \frac{-4}{8} = -\frac{1}{2} \)
- \( q = \frac{5(-\frac{1}{5}) + 5}{8} = \frac{-1 + 5}{8} = \frac{4}{8} = \frac{1}{2} \)

### Final Values:
- When \( a = 3 \), \( p = \frac{3}{2} \), \( q = \frac{5}{2} \)
- When \( a = -\frac{1}{5} \), \( p = -\frac{1}{2} \), \( q = \frac{1}{2} \)