To solve the quadratic equation [tex]\( x^2 - x - 1 = 0 \)[/tex], we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -1 \)[/tex].
Next, we calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \text{Discriminant} = (-1)^2 - 4 \cdot 1 \cdot (-1) = 1 + 4 = 5 \][/tex]
Now, we calculate the two solutions using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a} \][/tex]
Substituting the values:
[tex]\[ x_1 = \frac{-(-1) + \sqrt{5}}{2 \cdot 1} = \frac{1 + \sqrt{5}}{2} \approx 1.618033988749895 \][/tex]
[tex]\[ x_2 = \frac{-(-1) - \sqrt{5}}{2 \cdot 1} = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887498949 \][/tex]
Based on these calculations, the solutions are:
- The smaller solution [tex]\( x \)[/tex] is [tex]\( \boxed{-0.6180339887498949} \)[/tex].
- The larger solution [tex]\( x \)[/tex] is [tex]\( \boxed{1.618033988749895} \)[/tex].
Thus, we have found the solutions for the quadratic equation [tex]\( x^2 - x - 1 = 0 \)[/tex].
