Use the drop-down menus to complete the statements about the function p(x) = x(x – 1) + 1.

The value of a is
.

The value of b is
.

The value of c is
.

The value of the discriminant is
.

The quadratic function will intersect the x-axis
times.

1 answer

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.