Solve for y
y = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667
y=0
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3y^{2}-5y=0
Subtract 5y from both sides.
y\left(3y-5\right)=0
Factor out y.
y=0 y=\frac{5}{3}
To find equation solutions, solve y=0 and 3y-5=0.
3y^{2}-5y=0
Subtract 5y from both sides.
y=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-5\right)±5}{2\times 3}
Take the square root of \left(-5\right)^{2}.
y=\frac{5±5}{2\times 3}
The opposite of -5 is 5.
y=\frac{5±5}{6}
Multiply 2 times 3.
y=\frac{10}{6}
Now solve the equation y=\frac{5±5}{6} when ± is plus. Add 5 to 5.
y=\frac{5}{3}
Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.
y=\frac{0}{6}
Now solve the equation y=\frac{5±5}{6} when ± is minus. Subtract 5 from 5.
y=0
Divide 0 by 6.
y=\frac{5}{3} y=0
The equation is now solved.
3y^{2}-5y=0
Subtract 5y from both sides.
\frac{3y^{2}-5y}{3}=\frac{0}{3}
Divide both sides by 3.
y^{2}-\frac{5}{3}y=\frac{0}{3}
Dividing by 3 undoes the multiplication by 3.
y^{2}-\frac{5}{3}y=0
Divide 0 by 3.
y^{2}-\frac{5}{3}y+\left(-\frac{5}{6}\right)^{2}=\left(-\frac{5}{6}\right)^{2}
Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
y^{2}-\frac{5}{3}y+\frac{25}{36}=\frac{25}{36}
Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.
\left(y-\frac{5}{6}\right)^{2}=\frac{25}{36}
Factor y^{2}-\frac{5}{3}y+\frac{25}{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{5}{6}\right)^{2}}=\sqrt{\frac{25}{36}}
Take the square root of both sides of the equation.
y-\frac{5}{6}=\frac{5}{6} y-\frac{5}{6}=-\frac{5}{6}
Simplify.
y=\frac{5}{3} y=0
Add \frac{5}{6} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
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y = 3x + 4
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Matrix
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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}