What are the solutions to the equation φ^2-φ-1=0 ?

The solutions to the equation φ^2-φ-1=0 are φ=(1+sqrt(5))/2 ,φ=(1-sqrt(5))/2

Find the zeros of φ^2-φ-1=0

The zeros of φ^2-φ-1=0 are φ=(1+sqrt(5))/2 ,φ=(1-sqrt(5))/2

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φ^{2} - φ - 1 = 0

φ = \frac{1 + 5}{2}, φ = \frac{1 - 5}{2}

Decimal Notation

φ = 1.61803 \dots, φ = - 0.61803 \dots

Solution steps

φ^{2} - φ - 1 = 0

Solve with the quadratic formula

φ_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 1)^{2} - 4 \cdot 1 \cdot (- 1) = 5

φ_{1, 2} = \frac{- ( - 1 ) \pm 5}{2 \cdot 1}

Separate the solutions

φ_{1} = \frac{- ( - 1 ) + 5}{2 \cdot 1}, φ_{2} = \frac{- ( - 1 ) - 5}{2 \cdot 1}

φ = \frac{- ( - 1 ) + 5}{2 \cdot 1} : \frac{1 + 5}{2}

φ = \frac{- ( - 1 ) - 5}{2 \cdot 1} : \frac{1 - 5}{2}

The solutions to the quadratic equation are:

φ = \frac{1 + 5}{2}, φ = \frac{1 - 5}{2}

x, 2 x^{2} + y^{2} + 3 x y = 1

2 x^{2} - 24 x + 72 = - 162

y, x^{2} + y^{2} = 4

- x^{2} - x - 2 = 0

(x^{2} + 2 x + 2) = 0

What are the solutions to the equation φ^2-φ-1=0 ?
The solutions to the equation φ^2-φ-1=0 are φ=(1+sqrt(5))/2 ,φ=(1-sqrt(5))/2
Find the zeros of φ^2-φ-1=0
The zeros of φ^2-φ-1=0 are φ=(1+sqrt(5))/2 ,φ=(1-sqrt(5))/2