Solve 1/2-left(3-yright)^2=y-2x

$$\frac{ 1 }{ 2- { \left(3-y \right) }^{ 2 } } =y-2x$$

Solve for x

$$x=-\frac{y^{3}-6y^{2}+7y+1}{2\left(-y^{2}+6y-7\right)}$$

$$y\neq \sqrt{2}+3\text{ and }y\neq 3-\sqrt{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right).

$$-1=\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

To find the opposite of \sqrt{2}+3, find the opposite of each term.

$$-1=\left(y-\sqrt{2}-3\right)\left(y-\left(-\sqrt{2}+3\right)\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

To find the opposite of -\sqrt{2}+3, find the opposite of each term.

$$-1=\left(y-\sqrt{2}-3\right)\left(y+\sqrt{2}-3\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

Use the distributive property to multiply y-\sqrt{2}-3 by y+\sqrt{2}-3 and combine like terms.

$$-1=\left(y^{2}-6y-\left(\sqrt{2}\right)^{2}+9\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

The square of \sqrt{2} is 2.

$$-1=\left(y^{2}-6y-2+9\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

Add -2 and 9 to get 7.

$$-1=\left(y^{2}-6y+7\right)y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

Use the distributive property to multiply y^{2}-6y+7 by y.

$$-1=y^{3}-6y^{2}+7y-2x\left(y-\left(\sqrt{2}+3\right)\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

To find the opposite of \sqrt{2}+3, find the opposite of each term.

$$-1=y^{3}-6y^{2}+7y-2x\left(y-\sqrt{2}-3\right)\left(y-\left(-\sqrt{2}+3\right)\right)$$

To find the opposite of -\sqrt{2}+3, find the opposite of each term.

$$-1=y^{3}-6y^{2}+7y-2x\left(y-\sqrt{2}-3\right)\left(y+\sqrt{2}-3\right)$$

Use the distributive property to multiply -2x by y-\sqrt{2}-3.

$$-1=y^{3}-6y^{2}+7y+\left(-2xy+2\sqrt{2}x+6x\right)\left(y+\sqrt{2}-3\right)$$

Use the distributive property to multiply -2xy+2\sqrt{2}x+6x by y+\sqrt{2}-3 and combine like terms.

$$-1=y^{3}-6y^{2}+7y-2xy^{2}+12yx+2x\left(\sqrt{2}\right)^{2}-18x$$

The square of \sqrt{2} is 2.

$$-1=y^{3}-6y^{2}+7y-2xy^{2}+12yx+2x\times 2-18x$$

Multiply 2 and 2 to get 4.

$$-1=y^{3}-6y^{2}+7y-2xy^{2}+12yx+4x-18x$$

Combine 4x and -18x to get -14x.

$$-1=y^{3}-6y^{2}+7y-2xy^{2}+12yx-14x$$

Swap sides so that all variable terms are on the left hand side.

$$y^{3}-6y^{2}+7y-2xy^{2}+12yx-14x=-1$$

Subtract y^{3} from both sides.

$$-6y^{2}+7y-2xy^{2}+12yx-14x=-1-y^{3}$$

Add 6y^{2} to both sides.

$$7y-2xy^{2}+12yx-14x=-1-y^{3}+6y^{2}$$

Subtract 7y from both sides.

$$-2xy^{2}+12yx-14x=-1-y^{3}+6y^{2}-7y$$

Combine all terms containing x.

$$\left(-2y^{2}+12y-14\right)x=-1-y^{3}+6y^{2}-7y$$

The equation is in standard form.

$$\left(-2y^{2}+12y-14\right)x=-y^{3}+6y^{2}-7y-1$$

Divide both sides by -2y^{2}+12y-14.

$$\frac{\left(-2y^{2}+12y-14\right)x}{-2y^{2}+12y-14}=\frac{-y^{3}+6y^{2}-7y-1}{-2y^{2}+12y-14}$$

Dividing by -2y^{2}+12y-14 undoes the multiplication by -2y^{2}+12y-14.

$$x=\frac{-y^{3}+6y^{2}-7y-1}{-2y^{2}+12y-14}$$

Divide 6y^{2}-7y-y^{3}-1 by -2y^{2}+12y-14.

$$x=\frac{-y^{3}+6y^{2}-7y-1}{2\left(-y^{2}+6y-7\right)}$$