Question

Use the discriminant to determine how many and what kind of solutions the quadratic equation x^2-x=1
has.

Answer 1

To determine the number and type of solutions for the quadratic equation given by \( x^2 - x = 1 \), we first need to rewrite it in standard form, which is \( ax^2 + bx + c = 0 \).

Starting from the original equation:

\[
x^2 - x - 1 = 0
\]

In this equation, we identify the coefficients:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -1 \)

Next, we will calculate the discriminant \( D \) using the formula:

\[
D = b^2 - 4ac
\]

Substituting the values of \( a \), \( b \), and \( c \):

\[
D = (-1)^2 - 4(1)(-1) = 1 + 4 = 5
\]

Now, we analyze the value of the discriminant:

- If \( D > 0 \), the quadratic equation has two distinct real solutions.
- If \( D = 0 \), the quadratic equation has exactly one real solution (a repeated root).
- If \( D < 0 \), the quadratic equation has two complex solutions (no real solutions).

Since we have \( D = 5 \), which is greater than zero, we conclude that the quadratic equation \( x^2 - x - 1 = 0 \) has **two distinct real solutions**.