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
Graph
Share
Copied to clipboard
36y\left(-27\right)y=-27y\times 12+18
Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by -27y.
-972yy=-27y\times 12+18
Multiply 36 and -27 to get -972.
-972y^{2}=-27y\times 12+18
Multiply y and y to get y^{2}.
-972y^{2}=-324y+18
Multiply -27 and 12 to get -324.
-972y^{2}+324y=18
Add 324y to both sides.
-972y^{2}+324y-18=0
Subtract 18 from both sides.
y=\frac{-324±\sqrt{324^{2}-4\left(-972\right)\left(-18\right)}}{2\left(-972\right)}
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{104976-4\left(-972\right)\left(-18\right)}}{2\left(-972\right)}
Square 324.
y=\frac{-324±\sqrt{104976+3888\left(-18\right)}}{2\left(-972\right)}
Multiply -4 times -972.
y=\frac{-324±\sqrt{104976-69984}}{2\left(-972\right)}
Multiply 3888 times -18.
y=\frac{-324±\sqrt{34992}}{2\left(-972\right)}
Add 104976 to -69984.
y=\frac{-324±108\sqrt{3}}{2\left(-972\right)}
Take the square root of 34992.
y=\frac{-324±108\sqrt{3}}{-1944}
Multiply 2 times -972.
y=\frac{108\sqrt{3}-324}{-1944}
Now solve the equation y=\frac{-324±108\sqrt{3}}{-1944} when ± is plus. Add -324 to 108\sqrt{3}.
y=-\frac{\sqrt{3}}{18}+\frac{1}{6}
Divide -324+108\sqrt{3} by -1944.
y=\frac{-108\sqrt{3}-324}{-1944}
Now solve the equation y=\frac{-324±108\sqrt{3}}{-1944} when ± is minus. Subtract 108\sqrt{3} from -324.
y=\frac{\sqrt{3}}{18}+\frac{1}{6}
Divide -324-108\sqrt{3} by -1944.
y=-\frac{\sqrt{3}}{18}+\frac{1}{6} y=\frac{\sqrt{3}}{18}+\frac{1}{6}
The equation is now solved.
36y\left(-27\right)y=-27y\times 12+18
Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by -27y.
-972yy=-27y\times 12+18
Multiply 36 and -27 to get -972.
-972y^{2}=-27y\times 12+18
Multiply y and y to get y^{2}.
-972y^{2}=-324y+18
Multiply -27 and 12 to get -324.
-972y^{2}+324y=18
Add 324y to both sides.
\frac{-972y^{2}+324y}{-972}=\frac{18}{-972}
Divide both sides by -972.
y^{2}+\frac{324}{-972}y=\frac{18}{-972}
Dividing by -972 undoes the multiplication by -972.
y^{2}-\frac{1}{3}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{1}{54}
Reduce the fraction \frac{18}{-972} to lowest terms by extracting and canceling out 18.
y^{2}-\frac{1}{3}y+\left(-\frac{1}{6}\right)^{2}=-\frac{1}{54}+\left(-\frac{1}{6}\right)^{2}
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+\frac{1}{36}=-\frac{1}{54}+\frac{1}{36}
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}{108}
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.
\left(y-\frac{1}{6}\right)^{2}=\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}.
\sqrt{\left(y-\frac{1}{6}\right)^{2}}=\sqrt{\frac{1}{108}}
Take the square root of both sides of the equation.
y-\frac{1}{6}=\frac{\sqrt{3}}{18} y-\frac{1}{6}=-\frac{\sqrt{3}}{18}
Simplify.
y=\frac{\sqrt{3}}{18}+\frac{1}{6} y=-\frac{\sqrt{3}}{18}+\frac{1}{6}
Add \frac{1}{6} to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}