Solve 2(y^2-18y+81)=y

$$2 ( y ^ { 2 } - 18 y + 81 ) = y$$

Solve for y

$$y = \frac{\sqrt{73} + 37}{4} \approx 11.386000936$$

$$y = \frac{37 - \sqrt{73}}{4} \approx 7.113999064$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by y^{2}-18y+81.

$$2y^{2}-36y+162=y$$

Subtract y from both sides.

$$2y^{2}-36y+162-y=0$$

Combine -36y and -y to get -37y.

$$2y^{2}-37y+162=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -37 for b, and 162 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 2\times 162}}{2\times 2}$$

Square -37.

$$y=\frac{-\left(-37\right)±\sqrt{1369-4\times 2\times 162}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-37\right)±\sqrt{1369-8\times 162}}{2\times 2}$$

Multiply -8 times 162.

$$y=\frac{-\left(-37\right)±\sqrt{1369-1296}}{2\times 2}$$

Add 1369 to -1296.

$$y=\frac{-\left(-37\right)±\sqrt{73}}{2\times 2}$$

The opposite of -37 is 37.

$$y=\frac{37±\sqrt{73}}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{37±\sqrt{73}}{4}$$

Now solve the equation y=\frac{37±\sqrt{73}}{4} when ± is plus. Add 37 to \sqrt{73}.

$$y=\frac{\sqrt{73}+37}{4}$$

Now solve the equation y=\frac{37±\sqrt{73}}{4} when ± is minus. Subtract \sqrt{73} from 37.

$$y=\frac{37-\sqrt{73}}{4}$$

The equation is now solved.

$$y=\frac{\sqrt{73}+37}{4} y=\frac{37-\sqrt{73}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2 by y^{2}-18y+81.

$$2y^{2}-36y+162=y$$

Subtract y from both sides.

$$2y^{2}-36y+162-y=0$$

Combine -36y and -y to get -37y.

$$2y^{2}-37y+162=0$$

Subtract 162 from both sides. Anything subtracted from zero gives its negation.

$$2y^{2}-37y=-162$$

Divide both sides by 2.

$$\frac{2y^{2}-37y}{2}=-\frac{162}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y^{2}-\frac{37}{2}y=-\frac{162}{2}$$

Divide -162 by 2.

$$y^{2}-\frac{37}{2}y=-81$$

Divide -\frac{37}{2}, the coefficient of the x term, by 2 to get -\frac{37}{4}. Then add the square of -\frac{37}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{37}{2}y+\left(-\frac{37}{4}\right)^{2}=-81+\left(-\frac{37}{4}\right)^{2}$$

Square -\frac{37}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{37}{2}y+\frac{1369}{16}=-81+\frac{1369}{16}$$

Add -81 to \frac{1369}{16}.

$$y^{2}-\frac{37}{2}y+\frac{1369}{16}=\frac{73}{16}$$

Factor y^{2}-\frac{37}{2}y+\frac{1369}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{37}{4}\right)^{2}=\frac{73}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{37}{4}\right)^{2}}=\sqrt{\frac{73}{16}}$$

Simplify.

$$y-\frac{37}{4}=\frac{\sqrt{73}}{4} y-\frac{37}{4}=-\frac{\sqrt{73}}{4}$$

Add \frac{37}{4} to both sides of the equation.

$$y=\frac{\sqrt{73}+37}{4} y=\frac{37-\sqrt{73}}{4}$$