Solve 8y-5/2y+5=5y+7/-3y+2

$$\frac { 8 y - 5 } { 2 y + 5 } = 5 \frac { y + 7 } { - 3 y + 2 }$$

Solve for y

$$y=-\frac{\sqrt{5266}i}{34}-\frac{16}{17}\approx -0.941176471-2.134329713i$$

$$y=\frac{\sqrt{5266}i}{34}-\frac{16}{17}\approx -0.941176471+2.134329713i$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values -\frac{5}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3y-2\right)\left(2y+5\right), the least common multiple of 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(y+7\right)$$

Use the distributive property to multiply 3y-2 by 8y-5 and combine like terms.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(y+7\right)$$

Use the distributive property to multiply 5 by -5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(y+7\right)$$

Use the distributive property to multiply -25-10y by y+7 and combine like terms.

$$24y^{2}-31y+10=-95y-175-10y^{2}$$

Add 95y to both sides.

$$24y^{2}-31y+10+95y=-175-10y^{2}$$

Combine -31y and 95y to get 64y.

$$24y^{2}+64y+10=-175-10y^{2}$$

Subtract -175 from both sides.

$$24y^{2}+64y+10-\left(-175\right)=-10y^{2}$$

The opposite of -175 is 175.

$$24y^{2}+64y+10+175=-10y^{2}$$

Add 10y^{2} to both sides.

$$24y^{2}+64y+10+175+10y^{2}=0$$

Add 10 and 175 to get 185.

$$24y^{2}+64y+185+10y^{2}=0$$

Combine 24y^{2} and 10y^{2} to get 34y^{2}.

$$34y^{2}+64y+185=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 34 for a, 64 for b, and 185 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-64±\sqrt{64^{2}-4\times 34\times 185}}{2\times 34}$$

Square 64.

$$y=\frac{-64±\sqrt{4096-4\times 34\times 185}}{2\times 34}$$

Multiply -4 times 34.

$$y=\frac{-64±\sqrt{4096-136\times 185}}{2\times 34}$$

Multiply -136 times 185.

$$y=\frac{-64±\sqrt{4096-25160}}{2\times 34}$$

Add 4096 to -25160.

$$y=\frac{-64±\sqrt{-21064}}{2\times 34}$$

Take the square root of -21064.

$$y=\frac{-64±2\sqrt{5266}i}{2\times 34}$$

Multiply 2 times 34.

$$y=\frac{-64±2\sqrt{5266}i}{68}$$

Now solve the equation y=\frac{-64±2\sqrt{5266}i}{68} when ± is plus. Add -64 to 2i\sqrt{5266}.

$$y=\frac{-64+2\sqrt{5266}i}{68}$$

Divide -64+2i\sqrt{5266} by 68.

$$y=\frac{\sqrt{5266}i}{34}-\frac{16}{17}$$

Now solve the equation y=\frac{-64±2\sqrt{5266}i}{68} when ± is minus. Subtract 2i\sqrt{5266} from -64.

$$y=\frac{-2\sqrt{5266}i-64}{68}$$

Divide -64-2i\sqrt{5266} by 68.

$$y=-\frac{\sqrt{5266}i}{34}-\frac{16}{17}$$

The equation is now solved.

$$y=\frac{\sqrt{5266}i}{34}-\frac{16}{17} y=-\frac{\sqrt{5266}i}{34}-\frac{16}{17}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values -\frac{5}{2},\frac{2}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3y-2\right)\left(2y+5\right), the least common multiple of 2y+5,-3y+2.

$$\left(3y-2\right)\left(8y-5\right)=5\left(-5-2y\right)\left(y+7\right)$$

Use the distributive property to multiply 3y-2 by 8y-5 and combine like terms.

$$24y^{2}-31y+10=5\left(-5-2y\right)\left(y+7\right)$$

Use the distributive property to multiply 5 by -5-2y.

$$24y^{2}-31y+10=\left(-25-10y\right)\left(y+7\right)$$

Use the distributive property to multiply -25-10y by y+7 and combine like terms.

$$24y^{2}-31y+10=-95y-175-10y^{2}$$

Add 95y to both sides.

$$24y^{2}-31y+10+95y=-175-10y^{2}$$

Combine -31y and 95y to get 64y.

$$24y^{2}+64y+10=-175-10y^{2}$$

Add 10y^{2} to both sides.

$$24y^{2}+64y+10+10y^{2}=-175$$

Combine 24y^{2} and 10y^{2} to get 34y^{2}.

$$34y^{2}+64y+10=-175$$

Subtract 10 from both sides.

$$34y^{2}+64y=-175-10$$

Subtract 10 from -175 to get -185.

$$34y^{2}+64y=-185$$

Divide both sides by 34.

$$\frac{34y^{2}+64y}{34}=-\frac{185}{34}$$

Dividing by 34 undoes the multiplication by 34.

$$y^{2}+\frac{64}{34}y=-\frac{185}{34}$$

Reduce the fraction \frac{64}{34} to lowest terms by extracting and canceling out 2.

$$y^{2}+\frac{32}{17}y=-\frac{185}{34}$$

Divide \frac{32}{17}, the coefficient of the x term, by 2 to get \frac{16}{17}. Then add the square of \frac{16}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{32}{17}y+\left(\frac{16}{17}\right)^{2}=-\frac{185}{34}+\left(\frac{16}{17}\right)^{2}$$

Square \frac{16}{17} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{32}{17}y+\frac{256}{289}=-\frac{185}{34}+\frac{256}{289}$$

Add -\frac{185}{34} to \frac{256}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{32}{17}y+\frac{256}{289}=-\frac{2633}{578}$$

Factor y^{2}+\frac{32}{17}y+\frac{256}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{16}{17}\right)^{2}=-\frac{2633}{578}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{16}{17}\right)^{2}}=\sqrt{-\frac{2633}{578}}$$

Simplify.

$$y+\frac{16}{17}=\frac{\sqrt{5266}i}{34} y+\frac{16}{17}=-\frac{\sqrt{5266}i}{34}$$

Subtract \frac{16}{17} from both sides of the equation.

$$y=\frac{\sqrt{5266}i}{34}-\frac{16}{17} y=-\frac{\sqrt{5266}i}{34}-\frac{16}{17}$$