Solve for y
y=-\frac{1}{3}\approx -0.333333333
y=2
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\frac{3}{5}y^{2}-\frac{2}{5}=y
Divide each term of 3y^{2}-2 by 5 to get \frac{3}{5}y^{2}-\frac{2}{5}.
\frac{3}{5}y^{2}-\frac{2}{5}-y=0
Subtract y from both sides.
\frac{3}{5}y^{2}-y-\frac{2}{5}=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
y=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{3}{5}\left(-\frac{2}{5}\right)}}{2\times \frac{3}{5}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{5} for a, -1 for b, and -\frac{2}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
y=\frac{-\left(-1\right)±\sqrt{1-\frac{12}{5}\left(-\frac{2}{5}\right)}}{2\times \frac{3}{5}}
Multiply -4 times \frac{3}{5}.
y=\frac{-\left(-1\right)±\sqrt{1+\frac{24}{25}}}{2\times \frac{3}{5}}
Multiply -\frac{12}{5} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
y=\frac{-\left(-1\right)±\sqrt{\frac{49}{25}}}{2\times \frac{3}{5}}
Add 1 to \frac{24}{25}.
y=\frac{-\left(-1\right)±\frac{7}{5}}{2\times \frac{3}{5}}
Take the square root of \frac{49}{25}.
y=\frac{1±\frac{7}{5}}{2\times \frac{3}{5}}
The opposite of -1 is 1.
y=\frac{1±\frac{7}{5}}{\frac{6}{5}}
Multiply 2 times \frac{3}{5}.
y=\frac{\frac{12}{5}}{\frac{6}{5}}
Now solve the equation y=\frac{1±\frac{7}{5}}{\frac{6}{5}} when ± is plus. Add 1 to \frac{7}{5}.
y=2
Divide \frac{12}{5} by \frac{6}{5} by multiplying \frac{12}{5} by the reciprocal of \frac{6}{5}.
y=-\frac{\frac{2}{5}}{\frac{6}{5}}
Now solve the equation y=\frac{1±\frac{7}{5}}{\frac{6}{5}} when ± is minus. Subtract \frac{7}{5} from 1.
y=-\frac{1}{3}
Divide -\frac{2}{5} by \frac{6}{5} by multiplying -\frac{2}{5} by the reciprocal of \frac{6}{5}.
y=2 y=-\frac{1}{3}
The equation is now solved.
\frac{3}{5}y^{2}-\frac{2}{5}=y
Divide each term of 3y^{2}-2 by 5 to get \frac{3}{5}y^{2}-\frac{2}{5}.
\frac{3}{5}y^{2}-\frac{2}{5}-y=0
Subtract y from both sides.
\frac{3}{5}y^{2}-y=\frac{2}{5}
Add \frac{2}{5} to both sides. Anything plus zero gives itself.
\frac{\frac{3}{5}y^{2}-y}{\frac{3}{5}}=\frac{\frac{2}{5}}{\frac{3}{5}}
Divide both sides of the equation by \frac{3}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.
y^{2}+\left(-\frac{1}{\frac{3}{5}}\right)y=\frac{\frac{2}{5}}{\frac{3}{5}}
Dividing by \frac{3}{5} undoes the multiplication by \frac{3}{5}.
y^{2}-\frac{5}{3}y=\frac{\frac{2}{5}}{\frac{3}{5}}
Divide -1 by \frac{3}{5} by multiplying -1 by the reciprocal of \frac{3}{5}.
y^{2}-\frac{5}{3}y=\frac{2}{3}
Divide \frac{2}{5} by \frac{3}{5} by multiplying \frac{2}{5} by the reciprocal of \frac{3}{5}.
y^{2}-\frac{5}{3}y+\left(-\frac{5}{6}\right)^{2}=\frac{2}{3}+\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{2}{3}+\frac{25}{36}
Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.
y^{2}-\frac{5}{3}y+\frac{25}{36}=\frac{49}{36}
Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(y-\frac{5}{6}\right)^{2}=\frac{49}{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{49}{36}}
Take the square root of both sides of the equation.
y-\frac{5}{6}=\frac{7}{6} y-\frac{5}{6}=-\frac{7}{6}
Simplify.
y=2 y=-\frac{1}{3}
Add \frac{5}{6} to both sides of the equation.
Examples
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{ 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}