If `(1-p)` is a root of quadratic equation `x^2+p x+(1-p)=0,` then find its roots.

Explanation

To solve the problem, we need to find the roots of the quadratic equation \( x^2 + px + (1 - p) = 0 \) given that \( (1 - p) \) is a root of this equation.

### Step 1: Substitute the root into the equation
Since \( (1 - p) \) is a root, we can substitute \( x = 1 - p \) into the equation:

\[ (1 - p)^2 + p(1 - p) + (1 - p) = 0 \]

### Step 2: Expand the equation
Now, we will expand the left-hand side:

\[ (1 - p)^2 = 1 - 2p + p^2 \]
\[ p(1 - p) = p - p^2 \]
\[ 1 - p = 1 - p \]

Putting these together, we have:

\[ 1 - 2p + p^2 + p - p^2 + 1 - p = 0 \]

### Step 3: Simplify the equation
Now, we can simplify the equation:

\[ 1 - 2p + p^2 + p - p^2 + 1 - p = 0 \]
Combine like terms:

\[ (1 + 1) + (-2p + p - p) = 0 \]
\[ 2 - 2p = 0 \]

### Step 4: Solve for \( p \)
Now, we can solve for \( p \):

\[ 2 - 2p = 0 \implies 2p = 2 \implies p = 1 \]

### Step 5: Substitute \( p \) back into the original equation
Now that we have \( p = 1 \), we substitute this value back into the original quadratic equation:

\[ x^2 + 1x + (1 - 1) = 0 \]
This simplifies to:

\[ x^2 + x = 0 \]

### Step 6: Factor the equation
We can factor this equation:

\[ x(x + 1) = 0 \]

### Step 7: Find the roots
Setting each factor to zero gives us the roots:

\[ x = 0 \quad \text{or} \quad x + 1 = 0 \implies x = -1 \]

Thus, the roots of the quadratic equation are \( x = 0 \) and \( x = -1 \).

### Final Answer
The roots of the equation are \( x = 0 \) and \( x = -1 \).

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