Solve varphi=1+1/1+varphi

$$\varphi = 1 + \frac { 1 } { 1 + \varphi }$$

Solve for φ

$$\phi =\sqrt{2}\approx 1.414213562$$

$$\phi =-\sqrt{2}\approx -1.414213562$$

Steps by Finding Square Root

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+\phi }{1+\phi }.

$$\phi =\frac{1+\phi }{1+\phi }+\frac{1}{1+\phi }$$

Since \frac{1+\phi }{1+\phi } and \frac{1}{1+\phi } have the same denominator, add them by adding their numerators.

$$\phi =\frac{1+\phi +1}{1+\phi }$$

Combine like terms in 1+\phi +1.

$$\phi =\frac{2+\phi }{1+\phi }$$

Subtract \frac{2+\phi }{1+\phi } from both sides.

$$\phi -\frac{2+\phi }{1+\phi }=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \phi times \frac{1+\phi }{1+\phi }.

$$\frac{\phi \left(1+\phi \right)}{1+\phi }-\frac{2+\phi }{1+\phi }=0$$

Since \frac{\phi \left(1+\phi \right)}{1+\phi } and \frac{2+\phi }{1+\phi } have the same denominator, subtract them by subtracting their numerators.

$$\frac{\phi \left(1+\phi \right)-\left(2+\phi \right)}{1+\phi }=0$$

Do the multiplications in \phi \left(1+\phi \right)-\left(2+\phi \right).

$$\frac{\phi +\phi ^{2}-2-\phi }{1+\phi }=0$$

Combine like terms in \phi +\phi ^{2}-2-\phi .

$$\frac{\phi ^{2}-2}{1+\phi }=0$$

Variable \phi cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \phi +1.

$$\phi ^{2}-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$\phi ^{2}=2$$

Take the square root of both sides of the equation.

$$\phi =\sqrt{2} \phi =-\sqrt{2}$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{1+\phi }{1+\phi }.

$$\phi =\frac{1+\phi }{1+\phi }+\frac{1}{1+\phi }$$

Since \frac{1+\phi }{1+\phi } and \frac{1}{1+\phi } have the same denominator, add them by adding their numerators.

$$\phi =\frac{1+\phi +1}{1+\phi }$$

Combine like terms in 1+\phi +1.

$$\phi =\frac{2+\phi }{1+\phi }$$

Subtract \frac{2+\phi }{1+\phi } from both sides.

$$\phi -\frac{2+\phi }{1+\phi }=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \phi times \frac{1+\phi }{1+\phi }.

$$\frac{\phi \left(1+\phi \right)}{1+\phi }-\frac{2+\phi }{1+\phi }=0$$

Since \frac{\phi \left(1+\phi \right)}{1+\phi } and \frac{2+\phi }{1+\phi } have the same denominator, subtract them by subtracting their numerators.

$$\frac{\phi \left(1+\phi \right)-\left(2+\phi \right)}{1+\phi }=0$$

Do the multiplications in \phi \left(1+\phi \right)-\left(2+\phi \right).

$$\frac{\phi +\phi ^{2}-2-\phi }{1+\phi }=0$$

Combine like terms in \phi +\phi ^{2}-2-\phi .

$$\frac{\phi ^{2}-2}{1+\phi }=0$$

Variable \phi cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \phi +1.

$$\phi ^{2}-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\phi =\frac{0±\sqrt{0^{2}-4\left(-2\right)}}{2}$$

Square 0.

$$\phi =\frac{0±\sqrt{-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$\phi =\frac{0±\sqrt{8}}{2}$$

Take the square root of 8.

$$\phi =\frac{0±2\sqrt{2}}{2}$$

Now solve the equation \phi =\frac{0±2\sqrt{2}}{2} when ± is plus.

$$\phi =\sqrt{2}$$

Now solve the equation \phi =\frac{0±2\sqrt{2}}{2} when ± is minus.

$$\phi =-\sqrt{2}$$

The equation is now solved.

$$\phi =\sqrt{2} \phi =-\sqrt{2}$$