Solve varphi=1+1/1+frac{1{varphi}}

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

Solve for φ

$$\phi =\frac{\sqrt{5}+1}{2}\approx 1.618033989$$

$$\phi =\frac{1-\sqrt{5}}{2}\approx -0.618033989$$

Steps Using the Quadratic Formula

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

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

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

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

Variable \phi cannot be equal to 0 since division by zero is not defined. Divide 1 by \frac{\phi +1}{\phi } by multiplying 1 by the reciprocal of \frac{\phi +1}{\phi }.

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

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

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

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

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

Combine like terms in \phi +1+\phi .

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

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

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

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

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

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

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

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

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

Combine like terms in \phi ^{2}+\phi -2\phi -1.

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

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

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

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

$$\phi =\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$\phi =\frac{-\left(-1\right)±\sqrt{1+4}}{2}$$

Add 1 to 4.

$$\phi =\frac{-\left(-1\right)±\sqrt{5}}{2}$$

The opposite of -1 is 1.

$$\phi =\frac{1±\sqrt{5}}{2}$$

Now solve the equation \phi =\frac{1±\sqrt{5}}{2} when ± is plus. Add 1 to \sqrt{5}.

$$\phi =\frac{\sqrt{5}+1}{2}$$

Now solve the equation \phi =\frac{1±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from 1.

$$\phi =\frac{1-\sqrt{5}}{2}$$

The equation is now solved.

$$\phi =\frac{\sqrt{5}+1}{2} \phi =\frac{1-\sqrt{5}}{2}$$

Steps for Completing the Square

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

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

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

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

Variable \phi cannot be equal to 0 since division by zero is not defined. Divide 1 by \frac{\phi +1}{\phi } by multiplying 1 by the reciprocal of \frac{\phi +1}{\phi }.

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

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

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

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

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

Combine like terms in \phi +1+\phi .

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

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

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

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

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

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

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

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

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

Combine like terms in \phi ^{2}+\phi -2\phi -1.

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

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

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

Add 1 to both sides. Anything plus zero gives itself.

$$\phi ^{2}-\phi =1$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$\phi ^{2}-\phi +\frac{1}{4}=1+\frac{1}{4}$$

Add 1 to \frac{1}{4}.

$$\phi ^{2}-\phi +\frac{1}{4}=\frac{5}{4}$$

Factor \phi ^{2}-\phi +\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\phi -\frac{1}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\phi -\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$\phi -\frac{1}{2}=\frac{\sqrt{5}}{2} \phi -\frac{1}{2}=-\frac{\sqrt{5}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$\phi =\frac{\sqrt{5}+1}{2} \phi =\frac{1-\sqrt{5}}{2}$$