Solve 9y+27+24y+36/108*33y+63/108=-(y-1)/frac{-1{1-y}}

$$\frac { 9 y + 27 + 24 y + 36 } { 108 } \cdot \frac { 33 y + 63 } { 108 } = \frac { - ( y - 1 ) } { \frac { - 1 } { 1 - y } }$$

Solve for y

$$y=\frac{1065}{1417}+\frac{1152}{1417}i\approx 0.751587862+0.81298518i$$

$$y=\frac{1065}{1417}-\frac{1152}{1417}i\approx 0.751587862-0.81298518i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 108.

$$\left(9y+27+24y+36\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Combine 9y and 24y to get 33y.

$$\left(33y+27+36\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Add 27 and 36 to get 63.

$$\left(33y+63\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Express \left(33y+63\right)\times \frac{33y+63}{108} as a single fraction.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

To find the opposite of y-1, find the opposite of each term.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=\left(-108+108y\right)\left(-y+1\right)$$

Use the distributive property to multiply -108+108y by -y+1 and combine like terms.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=216y-108-108y^{2}$$

Multiply 33y+63 and 33y+63 to get \left(33y+63\right)^{2}.

$$\frac{\left(33y+63\right)^{2}}{108}=216y-108-108y^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(33y+63\right)^{2}.

$$\frac{1089y^{2}+4158y+3969}{108}=216y-108-108y^{2}$$

Divide each term of 1089y^{2}+4158y+3969 by 108 to get \frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}.

$$\frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}=216y-108-108y^{2}$$

Subtract 216y from both sides.

$$\frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}-216y=-108-108y^{2}$$

Combine \frac{77}{2}y and -216y to get -\frac{355}{2}y.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}=-108-108y^{2}$$

Subtract -108 from both sides.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}-\left(-108\right)=-108y^{2}$$

The opposite of -108 is 108.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}+108=-108y^{2}$$

Add 108y^{2} to both sides.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}+108+108y^{2}=0$$

Add \frac{147}{4} and 108 to get \frac{579}{4}.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{579}{4}+108y^{2}=0$$

Combine \frac{121}{12}y^{2} and 108y^{2} to get \frac{1417}{12}y^{2}.

$$\frac{1417}{12}y^{2}-\frac{355}{2}y+\frac{579}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1417}{12} for a, -\frac{355}{2} for b, and \frac{579}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{355}{2}\right)±\sqrt{\left(-\frac{355}{2}\right)^{2}-4\times \frac{1417}{12}\times \frac{579}{4}}}{2\times \frac{1417}{12}}$$

Square -\frac{355}{2} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-\frac{355}{2}\right)±\sqrt{\frac{126025}{4}-4\times \frac{1417}{12}\times \frac{579}{4}}}{2\times \frac{1417}{12}}$$

Multiply -4 times \frac{1417}{12}.

$$y=\frac{-\left(-\frac{355}{2}\right)±\sqrt{\frac{126025}{4}-\frac{1417}{3}\times \frac{579}{4}}}{2\times \frac{1417}{12}}$$

Multiply -\frac{1417}{3} times \frac{579}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{355}{2}\right)±\sqrt{\frac{126025-273481}{4}}}{2\times \frac{1417}{12}}$$

Add \frac{126025}{4} to -\frac{273481}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{355}{2}\right)±\sqrt{-36864}}{2\times \frac{1417}{12}}$$

Take the square root of -36864.

$$y=\frac{-\left(-\frac{355}{2}\right)±192i}{2\times \frac{1417}{12}}$$

The opposite of -\frac{355}{2} is \frac{355}{2}.

$$y=\frac{\frac{355}{2}±192i}{2\times \frac{1417}{12}}$$

Multiply 2 times \frac{1417}{12}.

$$y=\frac{\frac{355}{2}±192i}{\frac{1417}{6}}$$

Now solve the equation y=\frac{\frac{355}{2}±192i}{\frac{1417}{6}} when ± is plus. Add \frac{355}{2} to 192i.

$$y=\frac{\frac{355}{2}+192i}{\frac{1417}{6}}$$

Divide \frac{355}{2}+192i by \frac{1417}{6} by multiplying \frac{355}{2}+192i by the reciprocal of \frac{1417}{6}.

$$y=\frac{1065}{1417}+\frac{1152}{1417}i$$

Now solve the equation y=\frac{\frac{355}{2}±192i}{\frac{1417}{6}} when ± is minus. Subtract 192i from \frac{355}{2}.

$$y=\frac{\frac{355}{2}-192i}{\frac{1417}{6}}$$

Divide \frac{355}{2}-192i by \frac{1417}{6} by multiplying \frac{355}{2}-192i by the reciprocal of \frac{1417}{6}.

$$y=\frac{1065}{1417}-\frac{1152}{1417}i$$

The equation is now solved.

$$y=\frac{1065}{1417}+\frac{1152}{1417}i y=\frac{1065}{1417}-\frac{1152}{1417}i$$

Steps for Completing the Square

Multiply both sides of the equation by 108.

$$\left(9y+27+24y+36\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Combine 9y and 24y to get 33y.

$$\left(33y+27+36\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Add 27 and 36 to get 63.

$$\left(33y+63\right)\times \frac{33y+63}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

Express \left(33y+63\right)\times \frac{33y+63}{108} as a single fraction.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=\left(-108+108y\right)\left(-\left(y-1\right)\right)$$

To find the opposite of y-1, find the opposite of each term.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=\left(-108+108y\right)\left(-y+1\right)$$

Use the distributive property to multiply -108+108y by -y+1 and combine like terms.

$$\frac{\left(33y+63\right)\left(33y+63\right)}{108}=216y-108-108y^{2}$$

Multiply 33y+63 and 33y+63 to get \left(33y+63\right)^{2}.

$$\frac{\left(33y+63\right)^{2}}{108}=216y-108-108y^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(33y+63\right)^{2}.

$$\frac{1089y^{2}+4158y+3969}{108}=216y-108-108y^{2}$$

Divide each term of 1089y^{2}+4158y+3969 by 108 to get \frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}.

$$\frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}=216y-108-108y^{2}$$

Subtract 216y from both sides.

$$\frac{121}{12}y^{2}+\frac{77}{2}y+\frac{147}{4}-216y=-108-108y^{2}$$

Combine \frac{77}{2}y and -216y to get -\frac{355}{2}y.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}=-108-108y^{2}$$

Add 108y^{2} to both sides.

$$\frac{121}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}+108y^{2}=-108$$

Combine \frac{121}{12}y^{2} and 108y^{2} to get \frac{1417}{12}y^{2}.

$$\frac{1417}{12}y^{2}-\frac{355}{2}y+\frac{147}{4}=-108$$

Subtract \frac{147}{4} from both sides.

$$\frac{1417}{12}y^{2}-\frac{355}{2}y=-108-\frac{147}{4}$$

Subtract \frac{147}{4} from -108 to get -\frac{579}{4}.

$$\frac{1417}{12}y^{2}-\frac{355}{2}y=-\frac{579}{4}$$

Divide both sides of the equation by \frac{1417}{12}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{1417}{12}y^{2}-\frac{355}{2}y}{\frac{1417}{12}}=-\frac{\frac{579}{4}}{\frac{1417}{12}}$$

Dividing by \frac{1417}{12} undoes the multiplication by \frac{1417}{12}.

$$y^{2}+\left(-\frac{\frac{355}{2}}{\frac{1417}{12}}\right)y=-\frac{\frac{579}{4}}{\frac{1417}{12}}$$

Divide -\frac{355}{2} by \frac{1417}{12} by multiplying -\frac{355}{2} by the reciprocal of \frac{1417}{12}.

$$y^{2}-\frac{2130}{1417}y=-\frac{\frac{579}{4}}{\frac{1417}{12}}$$

Divide -\frac{579}{4} by \frac{1417}{12} by multiplying -\frac{579}{4} by the reciprocal of \frac{1417}{12}.

$$y^{2}-\frac{2130}{1417}y=-\frac{1737}{1417}$$

Divide -\frac{2130}{1417}, the coefficient of the x term, by 2 to get -\frac{1065}{1417}. Then add the square of -\frac{1065}{1417} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{2130}{1417}y+\left(-\frac{1065}{1417}\right)^{2}=-\frac{1737}{1417}+\left(-\frac{1065}{1417}\right)^{2}$$

Square -\frac{1065}{1417} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{2130}{1417}y+\frac{1134225}{2007889}=-\frac{1737}{1417}+\frac{1134225}{2007889}$$

Add -\frac{1737}{1417} to \frac{1134225}{2007889} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{2130}{1417}y+\frac{1134225}{2007889}=-\frac{1327104}{2007889}$$

Factor y^{2}-\frac{2130}{1417}y+\frac{1134225}{2007889}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{1065}{1417}\right)^{2}=-\frac{1327104}{2007889}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1065}{1417}\right)^{2}}=\sqrt{-\frac{1327104}{2007889}}$$

Simplify.

$$y-\frac{1065}{1417}=\frac{1152}{1417}i y-\frac{1065}{1417}=-\frac{1152}{1417}i$$

Add \frac{1065}{1417} to both sides of the equation.

$$y=\frac{1065}{1417}+\frac{1152}{1417}i y=\frac{1065}{1417}-\frac{1152}{1417}i$$