Solve 9y^2-90y+225+y^2+6y-30=-6y+5-0

$$9 y ^ { 2 } - 90 y + 225 + y ^ { 2 } + 6 y - 30 = - 6 y + 5 - 0$$

Solve for y

$$y=\frac{39+\sqrt{379}i}{10}\approx 3.9+1.946792233i$$

$$y=\frac{-\sqrt{379}i+39}{10}\approx 3.9-1.946792233i$$

Steps Using the Quadratic Formula

Combine 9y^{2} and y^{2} to get 10y^{2}.

$$10y^{2}-90y+225+6y-30=-6y+5-0$$

Combine -90y and 6y to get -84y.

$$10y^{2}-84y+225-30=-6y+5-0$$

Subtract 30 from 225 to get 195.

$$10y^{2}-84y+195=-6y+5-0$$

Add 0 to both sides.

$$10y^{2}-84y+195+0=-6y+5$$

Add 195 and 0 to get 195.

$$10y^{2}-84y+195=-6y+5$$

Add 6y to both sides.

$$10y^{2}-84y+195+6y=5$$

Combine -84y and 6y to get -78y.

$$10y^{2}-78y+195=5$$

Subtract 5 from both sides.

$$10y^{2}-78y+195-5=0$$

Subtract 5 from 195 to get 190.

$$10y^{2}-78y+190=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -78 for b, and 190 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-78\right)±\sqrt{\left(-78\right)^{2}-4\times 10\times 190}}{2\times 10}$$

Square -78.

$$y=\frac{-\left(-78\right)±\sqrt{6084-4\times 10\times 190}}{2\times 10}$$

Multiply -4 times 10.

$$y=\frac{-\left(-78\right)±\sqrt{6084-40\times 190}}{2\times 10}$$

Multiply -40 times 190.

$$y=\frac{-\left(-78\right)±\sqrt{6084-7600}}{2\times 10}$$

Add 6084 to -7600.

$$y=\frac{-\left(-78\right)±\sqrt{-1516}}{2\times 10}$$

Take the square root of -1516.

$$y=\frac{-\left(-78\right)±2\sqrt{379}i}{2\times 10}$$

The opposite of -78 is 78.

$$y=\frac{78±2\sqrt{379}i}{2\times 10}$$

Multiply 2 times 10.

$$y=\frac{78±2\sqrt{379}i}{20}$$

Now solve the equation y=\frac{78±2\sqrt{379}i}{20} when ± is plus. Add 78 to 2i\sqrt{379}.

$$y=\frac{78+2\sqrt{379}i}{20}$$

Divide 78+2i\sqrt{379} by 20.

$$y=\frac{39+\sqrt{379}i}{10}$$

Now solve the equation y=\frac{78±2\sqrt{379}i}{20} when ± is minus. Subtract 2i\sqrt{379} from 78.

$$y=\frac{-2\sqrt{379}i+78}{20}$$

Divide 78-2i\sqrt{379} by 20.

$$y=\frac{-\sqrt{379}i+39}{10}$$

The equation is now solved.

$$y=\frac{39+\sqrt{379}i}{10} y=\frac{-\sqrt{379}i+39}{10}$$

Steps for Completing the Square

Combine 9y^{2} and y^{2} to get 10y^{2}.

$$10y^{2}-90y+225+6y-30=-6y+5-0$$

Combine -90y and 6y to get -84y.

$$10y^{2}-84y+225-30=-6y+5-0$$

Subtract 30 from 225 to get 195.

$$10y^{2}-84y+195=-6y+5-0$$

Subtract -6y+5 from both sides.

$$10y^{2}-84y+195-\left(-6y+5\right)=-0$$

To find the opposite of -6y+5, find the opposite of each term.

$$10y^{2}-84y+195+6y-5=-0$$

Combine -84y and 6y to get -78y.

$$10y^{2}-78y+195-5=-0$$

Subtract 5 from 195 to get 190.

$$10y^{2}-78y+190=-0$$

Multiply -1 and 0 to get 0.

$$10y^{2}-78y+190=0$$

Subtract 190 from both sides. Anything subtracted from zero gives its negation.

$$10y^{2}-78y=-190$$

Divide both sides by 10.

$$\frac{10y^{2}-78y}{10}=-\frac{190}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$y^{2}+\left(-\frac{78}{10}\right)y=-\frac{190}{10}$$

Reduce the fraction \frac{-78}{10} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{39}{5}y=-\frac{190}{10}$$

Divide -190 by 10.

$$y^{2}-\frac{39}{5}y=-19$$

Divide -\frac{39}{5}, the coefficient of the x term, by 2 to get -\frac{39}{10}. Then add the square of -\frac{39}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{39}{5}y+\left(-\frac{39}{10}\right)^{2}=-19+\left(-\frac{39}{10}\right)^{2}$$

Square -\frac{39}{10} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{39}{5}y+\frac{1521}{100}=-19+\frac{1521}{100}$$

Add -19 to \frac{1521}{100}.

$$y^{2}-\frac{39}{5}y+\frac{1521}{100}=-\frac{379}{100}$$

Factor y^{2}-\frac{39}{5}y+\frac{1521}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{39}{10}\right)^{2}=-\frac{379}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{39}{10}\right)^{2}}=\sqrt{-\frac{379}{100}}$$

Simplify.

$$y-\frac{39}{10}=\frac{\sqrt{379}i}{10} y-\frac{39}{10}=-\frac{\sqrt{379}i}{10}$$

Add \frac{39}{10} to both sides of the equation.

$$y=\frac{39+\sqrt{379}i}{10} y=\frac{-\sqrt{379}i+39}{10}$$