Solve 36y=12+18:(-27y)

$$36 y = 12 + 18 : ( - 27 y )$$

Solve for y

$$y=\frac{\sqrt{3}}{18}+\frac{1}{6}\approx 0.262891712$$

$$y=-\frac{\sqrt{3}}{18}+\frac{1}{6}\approx 0.070441622$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by -27y.

$$36y\left(-27\right)y=-27y\times 12+18$$

Multiply 36 and -27 to get -972.

$$-972yy=-27y\times 12+18$$

Multiply y and y to get y^{2}.

$$-972y^{2}=-27y\times 12+18$$

Multiply -27 and 12 to get -324.

$$-972y^{2}=-324y+18$$

Add 324y to both sides.

$$-972y^{2}+324y=18$$

Subtract 18 from both sides.

$$-972y^{2}+324y-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -972 for a, 324 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-324±\sqrt{324^{2}-4\left(-972\right)\left(-18\right)}}{2\left(-972\right)}$$

Square 324.

$$y=\frac{-324±\sqrt{104976-4\left(-972\right)\left(-18\right)}}{2\left(-972\right)}$$

Multiply -4 times -972.

$$y=\frac{-324±\sqrt{104976+3888\left(-18\right)}}{2\left(-972\right)}$$

Multiply 3888 times -18.

$$y=\frac{-324±\sqrt{104976-69984}}{2\left(-972\right)}$$

Add 104976 to -69984.

$$y=\frac{-324±\sqrt{34992}}{2\left(-972\right)}$$

Take the square root of 34992.

$$y=\frac{-324±108\sqrt{3}}{2\left(-972\right)}$$

Multiply 2 times -972.

$$y=\frac{-324±108\sqrt{3}}{-1944}$$

Now solve the equation y=\frac{-324±108\sqrt{3}}{-1944} when ± is plus. Add -324 to 108\sqrt{3}.

$$y=\frac{108\sqrt{3}-324}{-1944}$$

Divide -324+108\sqrt{3} by -1944.

$$y=-\frac{\sqrt{3}}{18}+\frac{1}{6}$$

Now solve the equation y=\frac{-324±108\sqrt{3}}{-1944} when ± is minus. Subtract 108\sqrt{3} from -324.

$$y=\frac{-108\sqrt{3}-324}{-1944}$$

Divide -324-108\sqrt{3} by -1944.

$$y=\frac{\sqrt{3}}{18}+\frac{1}{6}$$

The equation is now solved.

$$y=-\frac{\sqrt{3}}{18}+\frac{1}{6} y=\frac{\sqrt{3}}{18}+\frac{1}{6}$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by -27y.

$$36y\left(-27\right)y=-27y\times 12+18$$

Multiply 36 and -27 to get -972.

$$-972yy=-27y\times 12+18$$

Multiply y and y to get y^{2}.

$$-972y^{2}=-27y\times 12+18$$

Multiply -27 and 12 to get -324.

$$-972y^{2}=-324y+18$$

Add 324y to both sides.

$$-972y^{2}+324y=18$$

Divide both sides by -972.

$$\frac{-972y^{2}+324y}{-972}=\frac{18}{-972}$$

Dividing by -972 undoes the multiplication by -972.

$$y^{2}+\frac{324}{-972}y=\frac{18}{-972}$$

Reduce the fraction \frac{324}{-972} to lowest terms by extracting and canceling out 324.

$$y^{2}-\frac{1}{3}y=\frac{18}{-972}$$

Reduce the fraction \frac{18}{-972} to lowest terms by extracting and canceling out 18.

$$y^{2}-\frac{1}{3}y=-\frac{1}{54}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{1}{3}y+\left(-\frac{1}{6}\right)^{2}=-\frac{1}{54}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=-\frac{1}{54}+\frac{1}{36}$$

Add -\frac{1}{54} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{1}{3}y+\frac{1}{36}=\frac{1}{108}$$

Factor y^{2}-\frac{1}{3}y+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{1}{6}\right)^{2}=\frac{1}{108}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{6}\right)^{2}}=\sqrt{\frac{1}{108}}$$

Simplify.

$$y-\frac{1}{6}=\frac{\sqrt{3}}{18} y-\frac{1}{6}=-\frac{\sqrt{3}}{18}$$

Add \frac{1}{6} to both sides of the equation.

$$y=\frac{\sqrt{3}}{18}+\frac{1}{6} y=-\frac{\sqrt{3}}{18}+\frac{1}{6}$$