Question

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.

Answer 1

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.