Solve for x xy^2+x^2y-6x-3=0

$x y^{2} + x^{2} y - 6 x - 3 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = y$ , $b = y^{2} - 6$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- (y^{2} - 6) \pm \sqrt{{(y^{2} - 6)}^{2} - 4 \cdot (y \cdot - 3)}}{2 y}$

Simplify the numerator.

Apply the distributive property.

$x = \frac{- y^{2} + 6 \pm \sqrt{{(y^{2} - 6)}^{2} - 4 \cdot y \cdot - 3}}{2 y}$

Multiply $- 1$ by $- 6$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{{(y^{2} - 6)}^{2} - 4 \cdot y \cdot - 3}}{2 y}$

Rewrite ${(y^{2} - 6)}^{2}$ as $(y^{2} - 6) (y^{2} - 6)$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{(y^{2} - 6) (y^{2} - 6) - 4 \cdot y \cdot - 3}}{2 y}$

Expand $(y^{2} - 6) (y^{2} - 6)$ using the FOIL Method.

Apply the distributive property.

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{2} (y^{2} - 6) - 6 (y^{2} - 6) - 4 \cdot y \cdot - 3}}{2 y}$

Apply the distributive property.

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{2} y^{2} + y^{2} \cdot - 6 - 6 (y^{2} - 6) - 4 \cdot y \cdot - 3}}{2 y}$

Apply the distributive property.

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{2} y^{2} + y^{2} \cdot - 6 - 6 y^{2} - 6 \cdot - 6 - 4 \cdot y \cdot - 3}}{2 y}$

Simplify and combine like terms.

Simplify each term.

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{2 + 2} + y^{2} \cdot - 6 - 6 y^{2} - 6 \cdot - 6 - 4 \cdot y \cdot - 3}}{2 y}$

Add $2$ and $2$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} + y^{2} \cdot - 6 - 6 y^{2} - 6 \cdot - 6 - 4 \cdot y \cdot - 3}}{2 y}$

Move $- 6$ to the left of $y^{2}$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} - 6 \cdot y^{2} - 6 y^{2} - 6 \cdot - 6 - 4 \cdot y \cdot - 3}}{2 y}$

Multiply $- 6$ by $- 6$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} - 6 y^{2} - 6 y^{2} + 36 - 4 \cdot y \cdot - 3}}{2 y}$

Subtract $6 y^{2}$ from $- 6 y^{2}$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} - 12 y^{2} + 36 - 4 \cdot y \cdot - 3}}{2 y}$

Multiply $- 3$ by $- 4$ .

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} - 12 y^{2} + 36 + 12 y}}{2 y}$

Reorder terms.

$x = \frac{- y^{2} + 6 \pm \sqrt{y^{4} - 12 y^{2} + 12 y + 36}}{2 y}$

The final answer is the combination of both solutions.

$x = - \frac{y^{2} - 6 - \sqrt{y^{4} - 12 y^{2} + 12 y + 36}}{2 y}$

$x = - \frac{y^{2} - 6 + \sqrt{y^{4} - 12 y^{2} + 12 y + 36}}{2 y}$