Solve 87.564=5/(1+y)^2+5/(1+y)+105/1

$$87.564 = \frac { 5 } { ( 1 + y ) ^ { 2 } } + \frac { 5 } { ( 1 + y ) } + \frac { 105 } { 1 }$$

Solve for y

$$y=\frac{-4984+i\times 25\sqrt{8093}}{4359}\approx -1.14338151+0.515950348i$$

$$y=\frac{-i\times 25\sqrt{8093}-4984}{4359}\approx -1.14338151-0.515950348i$$

Steps Using the Quadratic Formula

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

$$87.564\left(y+1\right)^{2}=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

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

$$87.564\left(y^{2}+2y+1\right)=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

Use the distributive property to multiply 87.564 by y^{2}+2y+1.

$$87.564y^{2}+175.128y+87.564=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

Use the distributive property to multiply y+1 by 5.

$$87.564y^{2}+175.128y+87.564=5+5y+5+\left(y+1\right)^{2}\times 105$$

Add 5 and 5 to get 10.

$$87.564y^{2}+175.128y+87.564=10+5y+\left(y+1\right)^{2}\times 105$$

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

$$87.564y^{2}+175.128y+87.564=10+5y+\left(y^{2}+2y+1\right)\times 105$$

Use the distributive property to multiply y^{2}+2y+1 by 105.

$$87.564y^{2}+175.128y+87.564=10+5y+105y^{2}+210y+105$$

Combine 5y and 210y to get 215y.

$$87.564y^{2}+175.128y+87.564=10+215y+105y^{2}+105$$

Add 10 and 105 to get 115.

$$87.564y^{2}+175.128y+87.564=115+215y+105y^{2}$$

Subtract 115 from both sides.

$$87.564y^{2}+175.128y+87.564-115=215y+105y^{2}$$

Subtract 115 from 87.564 to get -27.436.

$$87.564y^{2}+175.128y-27.436=215y+105y^{2}$$

Subtract 215y from both sides.

$$87.564y^{2}+175.128y-27.436-215y=105y^{2}$$

Combine 175.128y and -215y to get -39.872y.

$$87.564y^{2}-39.872y-27.436=105y^{2}$$

Subtract 105y^{2} from both sides.

$$87.564y^{2}-39.872y-27.436-105y^{2}=0$$

Combine 87.564y^{2} and -105y^{2} to get -17.436y^{2}.

$$-17.436y^{2}-39.872y-27.436=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-\frac{4359}{250}y^{2}-39.872y-27.436=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{4359}{250} for a, -39.872 for b, and -27.436 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-39.872\right)±\sqrt{\left(-39.872\right)^{2}-4\left(-\frac{4359}{250}\right)\left(-27.436\right)}}{2\left(-\frac{4359}{250}\right)}$$

Square -39.872 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-39.872\right)±\sqrt{1589.776384-4\left(-\frac{4359}{250}\right)\left(-27.436\right)}}{2\left(-\frac{4359}{250}\right)}$$

Multiply -4 times -\frac{4359}{250}.

$$y=\frac{-\left(-39.872\right)±\sqrt{1589.776384+\frac{8718}{125}\left(-27.436\right)}}{2\left(-\frac{4359}{250}\right)}$$

Multiply \frac{8718}{125} times -27.436 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-39.872\right)±\sqrt{\frac{24840256-29898381}{15625}}}{2\left(-\frac{4359}{250}\right)}$$

Add 1589.776384 to -\frac{29898381}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-39.872\right)±\sqrt{-\frac{8093}{25}}}{2\left(-\frac{4359}{250}\right)}$$

Take the square root of -\frac{8093}{25}.

$$y=\frac{-\left(-39.872\right)±\frac{\sqrt{8093}i}{5}}{2\left(-\frac{4359}{250}\right)}$$

The opposite of -39.872 is 39.872.

$$y=\frac{39.872±\frac{\sqrt{8093}i}{5}}{2\left(-\frac{4359}{250}\right)}$$

Multiply 2 times -\frac{4359}{250}.

$$y=\frac{39.872±\frac{\sqrt{8093}i}{5}}{-\frac{4359}{125}}$$

Now solve the equation y=\frac{39.872±\frac{\sqrt{8093}i}{5}}{-\frac{4359}{125}} when ± is plus. Add 39.872 to \frac{i\sqrt{8093}}{5}.

$$y=\frac{\frac{\sqrt{8093}i}{5}+\frac{4984}{125}}{-\frac{4359}{125}}$$

Divide \frac{4984}{125}+\frac{i\sqrt{8093}}{5} by -\frac{4359}{125} by multiplying \frac{4984}{125}+\frac{i\sqrt{8093}}{5} by the reciprocal of -\frac{4359}{125}.

$$y=\frac{-25\sqrt{8093}i-4984}{4359}$$

Now solve the equation y=\frac{39.872±\frac{\sqrt{8093}i}{5}}{-\frac{4359}{125}} when ± is minus. Subtract \frac{i\sqrt{8093}}{5} from 39.872.

$$y=\frac{-\frac{\sqrt{8093}i}{5}+\frac{4984}{125}}{-\frac{4359}{125}}$$

Divide \frac{4984}{125}-\frac{i\sqrt{8093}}{5} by -\frac{4359}{125} by multiplying \frac{4984}{125}-\frac{i\sqrt{8093}}{5} by the reciprocal of -\frac{4359}{125}.

$$y=\frac{-4984+25\sqrt{8093}i}{4359}$$

The equation is now solved.

$$y=\frac{-25\sqrt{8093}i-4984}{4359} y=\frac{-4984+25\sqrt{8093}i}{4359}$$

Steps for Completing the Square

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

$$87.564\left(y+1\right)^{2}=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

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

$$87.564\left(y^{2}+2y+1\right)=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

Use the distributive property to multiply 87.564 by y^{2}+2y+1.

$$87.564y^{2}+175.128y+87.564=5+\left(y+1\right)\times 5+\left(y+1\right)^{2}\times 105$$

Use the distributive property to multiply y+1 by 5.

$$87.564y^{2}+175.128y+87.564=5+5y+5+\left(y+1\right)^{2}\times 105$$

Add 5 and 5 to get 10.

$$87.564y^{2}+175.128y+87.564=10+5y+\left(y+1\right)^{2}\times 105$$

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

$$87.564y^{2}+175.128y+87.564=10+5y+\left(y^{2}+2y+1\right)\times 105$$

Use the distributive property to multiply y^{2}+2y+1 by 105.

$$87.564y^{2}+175.128y+87.564=10+5y+105y^{2}+210y+105$$

Combine 5y and 210y to get 215y.

$$87.564y^{2}+175.128y+87.564=10+215y+105y^{2}+105$$

Add 10 and 105 to get 115.

$$87.564y^{2}+175.128y+87.564=115+215y+105y^{2}$$

Subtract 215y from both sides.

$$87.564y^{2}+175.128y+87.564-215y=115+105y^{2}$$

Combine 175.128y and -215y to get -39.872y.

$$87.564y^{2}-39.872y+87.564=115+105y^{2}$$

Subtract 105y^{2} from both sides.

$$87.564y^{2}-39.872y+87.564-105y^{2}=115$$

Combine 87.564y^{2} and -105y^{2} to get -17.436y^{2}.

$$-17.436y^{2}-39.872y+87.564=115$$

Subtract 87.564 from both sides.

$$-17.436y^{2}-39.872y=115-87.564$$

Subtract 87.564 from 115 to get 27.436.

$$-17.436y^{2}-39.872y=27.436$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-\frac{4359}{250}y^{2}-39.872y=27.436$$

Divide both sides of the equation by -\frac{4359}{250}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{4359}{250}y^{2}-39.872y}{-\frac{4359}{250}}=\frac{27.436}{-\frac{4359}{250}}$$

Dividing by -\frac{4359}{250} undoes the multiplication by -\frac{4359}{250}.

$$y^{2}+\left(-\frac{39.872}{-\frac{4359}{250}}\right)y=\frac{27.436}{-\frac{4359}{250}}$$

Divide -39.872 by -\frac{4359}{250} by multiplying -39.872 by the reciprocal of -\frac{4359}{250}.

$$y^{2}+\frac{9968}{4359}y=\frac{27.436}{-\frac{4359}{250}}$$

Divide 27.436 by -\frac{4359}{250} by multiplying 27.436 by the reciprocal of -\frac{4359}{250}.

$$y^{2}+\frac{9968}{4359}y=-\frac{6859}{4359}$$

Divide \frac{9968}{4359}, the coefficient of the x term, by 2 to get \frac{4984}{4359}. Then add the square of \frac{4984}{4359} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{9968}{4359}y+\left(\frac{4984}{4359}\right)^{2}=-\frac{6859}{4359}+\left(\frac{4984}{4359}\right)^{2}$$

Square \frac{4984}{4359} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{9968}{4359}y+\frac{24840256}{19000881}=-\frac{6859}{4359}+\frac{24840256}{19000881}$$

Add -\frac{6859}{4359} to \frac{24840256}{19000881} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{9968}{4359}y+\frac{24840256}{19000881}=-\frac{5058125}{19000881}$$

Factor y^{2}+\frac{9968}{4359}y+\frac{24840256}{19000881}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{4984}{4359}\right)^{2}=-\frac{5058125}{19000881}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{4984}{4359}\right)^{2}}=\sqrt{-\frac{5058125}{19000881}}$$

Simplify.

$$y+\frac{4984}{4359}=\frac{25\sqrt{8093}i}{4359} y+\frac{4984}{4359}=-\frac{25\sqrt{8093}i}{4359}$$

Subtract \frac{4984}{4359} from both sides of the equation.

$$y=\frac{-4984+25\sqrt{8093}i}{4359} y=\frac{-25\sqrt{8093}i-4984}{4359}$$