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.
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