If `(1-p)` is a root of the quadratic equation `x^(2)+px+(1-p)=0`, then its roots are
(a) 0,-1
(b) -1,1
(c) 0,1
(d) -1,2

Answer

To solve the problem step by step, we start with the given quadratic equation and the information provided about its root.

### Step 1: Write down the given quadratic equation
The quadratic equation given is:
\[ x^2 + px + (1 - p) = 0 \]

### Step 2: Substitute the root into the equation
We know that \( (1 - p) \) is a root of the equation. Therefore, we substitute \( x = 1 - p \) into the equation:
\[ (1 - p)^2 + p(1 - p) + (1 - p) = 0 \]

### Step 3: Expand and simplify the equation
Now, we will expand and simplify the equation:
1. Expand \( (1 - p)^2 \):
\[ (1 - p)^2 = 1 - 2p + p^2 \]
2. Expand \( p(1 - p) \):
\[ p(1 - p) = p - p^2 \]
3. Substitute these back into the equation:
\[ 1 - 2p + p^2 + p - p^2 + 1 - p = 0 \]

Now, combine like terms:
\[ 1 + 1 - 2p + p - p + p^2 - p^2 = 0 \]
This simplifies to:
\[ 2 - 2p = 0 \]

### Step 4: Solve for \( p \)
Now, we can solve for \( p \):
\[ 2 - 2p = 0 \]
\[ 2p = 2 \]
\[ p = 1 \]

### Step 5: Substitute \( p \) back into the original equation
Now that we have \( p = 1 \), we substitute it back into the original quadratic equation:
\[ x^2 + 1x + (1 - 1) = 0 \]
This simplifies to:
\[ x^2 + x = 0 \]

### Step 6: Factor the equation
Now we can factor the equation:
\[ x(x + 1) = 0 \]

### Step 7: Find the roots
Setting each factor to zero gives us the roots:
1. \( x = 0 \)
2. \( x + 1 = 0 \) which gives \( x = -1 \)

### Conclusion
The roots of the quadratic equation are \( 0 \) and \( -1 \). Therefore, the correct option is:
**(a) 0, -1**