To analyze the function \( p(x) = x(x - 1) + 1 \), let's first rewrite it in standard quadratic form \( ax^2 + bx + c \).
Starting from \( p(x) \):
\[ p(x) = x(x - 1) + 1 \] \[ = x^2 - x + 1 \]
Now we can identify the coefficients:
- The coefficient \( a \) (of \( x^2 \)) is \( 1 \).
- The coefficient \( b \) (of \( x \)) is \( -1 \).
- The constant term \( c \) is \( 1 \).
Next, we calculate the discriminant (\( D \)) of the quadratic function, which is given by the formula:
\[ D = b^2 - 4ac \]
Substituting in the values we found:
\[ D = (-1)^2 - 4(1)(1) \] \[ = 1 - 4 \] \[ = -3 \]
The value of the discriminant is negative, which indicates that the quadratic function does not intersect the x-axis.
Now, to summarize:
- The value of \( a \) is 1.
- The value of \( b \) is -1.
- The value of \( c \) is 1.
- The value of the discriminant is -3.
- The quadratic function will intersect the x-axis 0 times.
Here are the completed statements:
The value of a is 1.
The value of b is -1.
The value of c is 1.
The value of the discriminant is -3.
The quadratic function will intersect the x-axis 0 times.