Step by step solution to "Quadratic substituted into itself : wrong?Let x2=x+1 the..."

Lesnaoq73

Answered question

2022-03-31

Quadratic substituted into itself : wrong?
Let

x^{2} = x + 1

the solution is the golden ratio phi :

x = \frac{1 \pm \sqrt{5}}{2}

horieblersee275

Beginner2022-04-01Added 17 answers

Step 1
When you square

M = W

M^{2} = W^{2}

you add in the extraneous solutions of

M = - W

x = (x^{2} - 1)

has two solutions. And

x^{2} = {(x^{2} - 1)}^{2}

will have 4 solutions: the two solutions to

x = x^{2} - 1

as well as two new and extraneous solutions to

x = - (x^{2} - 1)

x = \sqrt{\frac{3 \pm \sqrt{5}}{2}} = \frac{1 \pm \sqrt{5}}{2}

are the two solutions to

x = x^{2} - 1

x = - \sqrt{\frac{3 \pm \sqrt{5}}{2}} = - \frac{1 \pm \sqrt{5}}{2}

are the two solutions to

x = - (x^{2} - 1)

Notice that

{(\frac{1 \pm \sqrt{5}}{2})}^{2} = \frac{3 \pm \sqrt{5}}{2}

\sqrt{\frac{3 \pm \sqrt{5}}{2}} = \frac{1 \pm \sqrt{5}}{2}

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