The number of common chords of the parabola `y=x^(2)-x and x=y^(2)-y` is

Answer

To find the number of common chords of the parabolas \( y = x^2 - x \) and \( x = y^2 - y \), we will follow these steps:

### Step 1: Set the equations equal to each other
We have two parabolas:
1. \( y = x^2 - x \)
2. \( x = y^2 - y \)

To find the points of intersection, we can substitute \( y \) from the first equation into the second equation.

### Step 2: Substitute \( y \) into the second equation
Substituting \( y = x^2 - x \) into \( x = y^2 - y \):
\[ x = (x^2 - x)^2 - (x^2 - x) \]

### Step 3: Simplify the equation
Now, let's simplify the equation:
\[ x = (x^2 - x)^2 - (x^2 - x) \]
Expanding \( (x^2 - x)^2 \):
\[ = x^4 - 2x^3 + x^2 \]
So, we rewrite the equation:
\[ x = x^4 - 2x^3 + x^2 - x^2 + x \]
This simplifies to:
\[ x = x^4 - 2x^3 \]
Rearranging gives:
\[ x^4 - 2x^3 - x = 0 \]

### Step 4: Factor the equation
Factoring out \( x \):
\[ x(x^3 - 2x^2 - 1) = 0 \]
Thus, we have:
1. \( x = 0 \)
2. \( x^3 - 2x^2 - 1 = 0 \)

### Step 5: Solve the cubic equation
Now we need to solve the cubic equation \( x^3 - 2x^2 - 1 = 0 \). We can use numerical methods or graphing to find the roots of this cubic polynomial.

### Step 6: Find the number of real roots
Using the Rational Root Theorem or numerical methods, we find that this cubic equation has one real root (approximately \( x \approx 2.879 \)) and two complex roots.

### Step 7: Find corresponding \( y \) values
For each \( x \) value we found, we can find the corresponding \( y \) values using the equation \( y = x^2 - x \):
1. For \( x = 0 \): \( y = 0^2 - 0 = 0 \) (Point: (0, 0))
2. For \( x \approx 2.879 \): Calculate \( y \) using \( y = (2.879)^2 - 2.879 \).

### Step 8: Determine the number of common chords
The points of intersection are \( (0, 0) \) and the point corresponding to \( x \approx 2.879 \). Since there are two distinct intersection points, we can draw one common chord between these two points.

### Conclusion
Thus, the number of common chords of the given parabolas is **1**.

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