solvefor y,(x^3)/3+(y^2)/2-3x+2y=-1/6

Solution

solve for

y, \frac{x ^{3}}{3} + \frac{y ^{2}}{2} - 3 x + 2 y = - \frac{1}{6}

y = \frac{- 12 + - 24 x ^{3} + 216 x + 132}{6}, y = \frac{- 12 - - 24 x ^{3} + 216 x + 132}{6}

Solution steps

\frac{x ^{3}}{3} + \frac{y ^{2}}{2} - 3 x + 2 y = - \frac{1}{6}

Find Least Common Multiplier of

3, 2, 6 : 6

Multiply by LCM=

6

\frac{x ^{3}}{3} \cdot 6 + \frac{y ^{2}}{2} \cdot 6 - 3 x \cdot 6 + 2 y \cdot 6 = - \frac{1}{6} \cdot 6

Simplify

2 x^{3} + 3 y^{2} - 18 x + 12 y = - 1

Move

1

to the left side

2 x^{3} + 3 y^{2} - 18 x + 12 y + 1 = 0

Write in the standard form

a x^{2} + b x + c = 0

3 y^{2} + 12 y + 2 x^{3} - 18 x + 1 = 0

Solve with the quadratic formula

y_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 \cdot 3 ( 2 x ^{3} - 18 x + 1 )}{2 \cdot 3}

Simplify

1 2^{2} - 4 \cdot 3 (2 x^{3} - 18 x + 1) : - 24 x^{3} + 216 x + 132

y_{1, 2} = \frac{- 12 \pm - 24 x ^{3} + 216 x + 132}{2 \cdot 3}

Separate the solutions

y_{1} = \frac{- 12 + - 24 x ^{3} + 216 x + 132}{2 \cdot 3}, y_{2} = \frac{- 12 - - 24 x ^{3} + 216 x + 132}{2 \cdot 3}

y = \frac{- 12 + - 24 x ^{3} + 216 x + 132}{2 \cdot 3} : \frac{- 12 + - 24 x ^{3} + 216 x + 132}{6}

y = \frac{- 12 - - 24 x ^{3} + 216 x + 132}{2 \cdot 3} : \frac{- 12 - - 24 x ^{3} + 216 x + 132}{6}

The solutions to the quadratic equation are:

y = \frac{- 12 + - 24 x ^{3} + 216 x + 132}{6}, y = \frac{- 12 - - 24 x ^{3} + 216 x + 132}{6}

- 3 x^{2} - 6 x - 6 = 0

4 x^{2} - 8 x - 4 = 0

x^{2} + 4 x + 85 = 0

x^{2} - 6 x = - x^{2} + 6 x

a, 9 a^{2} + 12 ab = 32 b^{2}