Solve 1/(1+x)^2=36

$$\frac { 1 } { ( 1 + x ) ^ { 2 } } = 36$$

Solve for x

$$x = -\frac{7}{6} = -1\frac{1}{6} \approx -1.166666667$$

$$x=-\frac{5}{6}\approx -0.833333333$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}.

$$1=36\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$1=36\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 36 by x^{2}+2x+1.

$$1=36x^{2}+72x+36$$

Swap sides so that all variable terms are on the left hand side.

$$36x^{2}+72x+36=1$$

Subtract 1 from both sides.

$$36x^{2}+72x+36-1=0$$

Subtract 1 from 36 to get 35.

$$36x^{2}+72x+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 72 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\times 36\times 35}}{2\times 36}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\times 36\times 35}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-72±\sqrt{5184-144\times 35}}{2\times 36}$$

Multiply -144 times 35.

$$x=\frac{-72±\sqrt{5184-5040}}{2\times 36}$$

Add 5184 to -5040.

$$x=\frac{-72±\sqrt{144}}{2\times 36}$$

Take the square root of 144.

$$x=\frac{-72±12}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{-72±12}{72}$$

Now solve the equation x=\frac{-72±12}{72} when ± is plus. Add -72 to 12.

$$x=-\frac{60}{72}$$

Reduce the fraction \frac{-60}{72} to lowest terms by extracting and canceling out 12.

$$x=-\frac{5}{6}$$

Now solve the equation x=\frac{-72±12}{72} when ± is minus. Subtract 12 from -72.

$$x=-\frac{84}{72}$$

Reduce the fraction \frac{-84}{72} to lowest terms by extracting and canceling out 12.

$$x=-\frac{7}{6}$$

The equation is now solved.

$$x=-\frac{5}{6} x=-\frac{7}{6}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}.

$$1=36\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$1=36\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 36 by x^{2}+2x+1.

$$1=36x^{2}+72x+36$$

Swap sides so that all variable terms are on the left hand side.

$$36x^{2}+72x+36=1$$

Subtract 36 from both sides.

$$36x^{2}+72x=1-36$$

Subtract 36 from 1 to get -35.

$$36x^{2}+72x=-35$$

Divide both sides by 36.

$$\frac{36x^{2}+72x}{36}=-\frac{35}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\frac{72}{36}x=-\frac{35}{36}$$

Divide 72 by 36.

$$x^{2}+2x=-\frac{35}{36}$$

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

$$x^{2}+2x+1^{2}=-\frac{35}{36}+1^{2}$$

Square 1.

$$x^{2}+2x+1=-\frac{35}{36}+1$$

Add -\frac{35}{36} to 1.

$$x^{2}+2x+1=\frac{1}{36}$$

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

$$\left(x+1\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$x+1=\frac{1}{6} x+1=-\frac{1}{6}$$

Subtract 1 from both sides of the equation.

$$x=-\frac{5}{6} x=-\frac{7}{6}$$