Solve for [tex]$x$[/tex] using the asterisk method, if possible:

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

Be sure to enter the smaller solution first.

The smaller solution is [tex]x =[/tex] [tex]$\square$[/tex] .

The larger solution is [tex]x =[/tex] [tex]$\square$[/tex] .

Enter [tex]$N$[/tex] in the second blank if there is only one solution. Enter [tex]$N$[/tex] in both blanks if there is no rational solution.



Answer :

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].