Solve 97.379=2/1+x+102/(1+x)^2

$$97.379 = \frac { 2 } { 1 + x } + \frac { 102 } { ( 1 + x ) ^ { 2 } }$$

Solve for x

$$x=\frac{-20\sqrt{24834145}-96379}{97379}\approx -2.01323425$$

$$x=\frac{20\sqrt{24834145}-96379}{97379}\approx 0.033772559$$

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}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$97.379\left(x+1\right)^{2}=\left(x+1\right)\times 2+102$$

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

$$97.379\left(x^{2}+2x+1\right)=\left(x+1\right)\times 2+102$$

Use the distributive property to multiply 97.379 by x^{2}+2x+1.

$$97.379x^{2}+194.758x+97.379=\left(x+1\right)\times 2+102$$

Use the distributive property to multiply x+1 by 2.

$$97.379x^{2}+194.758x+97.379=2x+2+102$$

Add 2 and 102 to get 104.

$$97.379x^{2}+194.758x+97.379=2x+104$$

Subtract 2x from both sides.

$$97.379x^{2}+194.758x+97.379-2x=104$$

Combine 194.758x and -2x to get 192.758x.

$$97.379x^{2}+192.758x+97.379=104$$

Subtract 104 from both sides.

$$97.379x^{2}+192.758x+97.379-104=0$$

Subtract 104 from 97.379 to get -6.621.

$$97.379x^{2}+192.758x-6.621=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 97.379 for a, 192.758 for b, and -6.621 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192.758±\sqrt{192.758^{2}-4\times 97.379\left(-6.621\right)}}{2\times 97.379}$$

Square 192.758 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-192.758±\sqrt{37155.646564-4\times 97.379\left(-6.621\right)}}{2\times 97.379}$$

Multiply -4 times 97.379.

$$x=\frac{-192.758±\sqrt{37155.646564-389.516\left(-6.621\right)}}{2\times 97.379}$$

Multiply -389.516 times -6.621 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-192.758±\sqrt{\frac{9288911641+644746359}{250000}}}{2\times 97.379}$$

Add 37155.646564 to 2578.985436 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-192.758±\sqrt{39734.632}}{2\times 97.379}$$

Take the square root of 39734.632.

$$x=\frac{-192.758±\frac{\sqrt{24834145}}{25}}{2\times 97.379}$$

Multiply 2 times 97.379.

$$x=\frac{-192.758±\frac{\sqrt{24834145}}{25}}{194.758}$$

Now solve the equation x=\frac{-192.758±\frac{\sqrt{24834145}}{25}}{194.758} when ± is plus. Add -192.758 to \frac{\sqrt{24834145}}{25}.

$$x=\frac{\frac{\sqrt{24834145}}{25}-\frac{96379}{500}}{194.758}$$

Divide -\frac{96379}{500}+\frac{\sqrt{24834145}}{25} by 194.758 by multiplying -\frac{96379}{500}+\frac{\sqrt{24834145}}{25} by the reciprocal of 194.758.

$$x=\frac{20\sqrt{24834145}-96379}{97379}$$

Now solve the equation x=\frac{-192.758±\frac{\sqrt{24834145}}{25}}{194.758} when ± is minus. Subtract \frac{\sqrt{24834145}}{25} from -192.758.

$$x=\frac{-\frac{\sqrt{24834145}}{25}-\frac{96379}{500}}{194.758}$$

Divide -\frac{96379}{500}-\frac{\sqrt{24834145}}{25} by 194.758 by multiplying -\frac{96379}{500}-\frac{\sqrt{24834145}}{25} by the reciprocal of 194.758.

$$x=\frac{-20\sqrt{24834145}-96379}{97379}$$

The equation is now solved.

$$x=\frac{20\sqrt{24834145}-96379}{97379} x=\frac{-20\sqrt{24834145}-96379}{97379}$$

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}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$97.379\left(x+1\right)^{2}=\left(x+1\right)\times 2+102$$

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

$$97.379\left(x^{2}+2x+1\right)=\left(x+1\right)\times 2+102$$

Use the distributive property to multiply 97.379 by x^{2}+2x+1.

$$97.379x^{2}+194.758x+97.379=\left(x+1\right)\times 2+102$$

Use the distributive property to multiply x+1 by 2.

$$97.379x^{2}+194.758x+97.379=2x+2+102$$

Add 2 and 102 to get 104.

$$97.379x^{2}+194.758x+97.379=2x+104$$

Subtract 2x from both sides.

$$97.379x^{2}+194.758x+97.379-2x=104$$

Combine 194.758x and -2x to get 192.758x.

$$97.379x^{2}+192.758x+97.379=104$$

Subtract 97.379 from both sides.

$$97.379x^{2}+192.758x=104-97.379$$

Subtract 97.379 from 104 to get 6.621.

$$97.379x^{2}+192.758x=6.621$$

Divide both sides of the equation by 97.379, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{97.379x^{2}+192.758x}{97.379}=\frac{6.621}{97.379}$$

Dividing by 97.379 undoes the multiplication by 97.379.

$$x^{2}+\frac{192.758}{97.379}x=\frac{6.621}{97.379}$$

Divide 192.758 by 97.379 by multiplying 192.758 by the reciprocal of 97.379.

$$x^{2}+\frac{192758}{97379}x=\frac{6.621}{97.379}$$

Divide 6.621 by 97.379 by multiplying 6.621 by the reciprocal of 97.379.

$$x^{2}+\frac{192758}{97379}x=\frac{6621}{97379}$$

Divide \frac{192758}{97379}, the coefficient of the x term, by 2 to get \frac{96379}{97379}. Then add the square of \frac{96379}{97379} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{192758}{97379}x+\frac{96379}{97379}^{2}=\frac{6621}{97379}+\frac{96379}{97379}^{2}$$

Square \frac{96379}{97379} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{192758}{97379}x+\frac{9288911641}{9482669641}=\frac{6621}{97379}+\frac{9288911641}{9482669641}$$

Add \frac{6621}{97379} to \frac{9288911641}{9482669641} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{192758}{97379}x+\frac{9288911641}{9482669641}=\frac{9933658000}{9482669641}$$

Factor x^{2}+\frac{192758}{97379}x+\frac{9288911641}{9482669641}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{96379}{97379}\right)^{2}=\frac{9933658000}{9482669641}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{96379}{97379}\right)^{2}}=\sqrt{\frac{9933658000}{9482669641}}$$

Simplify.

$$x+\frac{96379}{97379}=\frac{20\sqrt{24834145}}{97379} x+\frac{96379}{97379}=-\frac{20\sqrt{24834145}}{97379}$$

Subtract \frac{96379}{97379} from both sides of the equation.

$$x=\frac{20\sqrt{24834145}-96379}{97379} x=\frac{-20\sqrt{24834145}-96379}{97379}$$