Solve y^2+1/y=2

$$y ^ { 2 } + \frac { 1 } { y } = 2$$

Solve for y

$$y=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

$$y=1$$

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

Solution Steps

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$yy^{2}+1=2y$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$y^{3}+1=2y$$

Subtract 2y from both sides.

$$y^{3}+1-2y=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$y^{3}-2y+1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$y=1$$

By Factor theorem, y-k is a factor of the polynomial for each root k. Divide y^{3}-2y+1 by y-1 to get y^{2}+y-1. Solve the equation where the result equals to 0.

$$y^{2}+y-1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$y=\frac{-1±\sqrt{5}}{2}$$

Solve the equation y^{2}+y-1=0 when ± is plus and when ± is minus.

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

List all found solutions.

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