Solve frac{(sqrt{x-y^2})^2+(sqrt{y+x^2})^2}{x+y/y-x}=

$$\frac { ( \sqrt { x - y ^ { 2 } } ) ^ { 2 } + ( \sqrt { y + x ^ { 2 } } ) ^ { 2 } } { \frac { x + y } { y - x } } =$$

Evaluate

$$\left(y-x\right)\left(x-y+1\right)$$

Short Solution Steps

Calculate \sqrt{x-y^{2}} to the power of 2 and get x-y^{2}.

$$\frac{x-y^{2}+\left(\sqrt{y+x^{2}}\right)^{2}}{\frac{x+y}{y-x}}$$

Calculate \sqrt{y+x^{2}} to the power of 2 and get y+x^{2}.

$$\frac{x-y^{2}+y+x^{2}}{\frac{x+y}{y-x}}$$

Divide x-y^{2}+y+x^{2} by \frac{x+y}{y-x} by multiplying x-y^{2}+y+x^{2} by the reciprocal of \frac{x+y}{y-x}.

$$\frac{\left(x-y^{2}+y+x^{2}\right)\left(y-x\right)}{x+y}$$

Factor the expressions that are not already factored.

$$\frac{\left(x+y\right)\left(-x+y\right)\left(x-y+1\right)}{x+y}$$

Cancel out x+y in both numerator and denominator.

$$\left(-x+y\right)\left(x-y+1\right)$$

Expand the expression.

$$-x^{2}+2xy-x-y^{2}+y$$

Factor

$$\left(y-x\right)\left(x-y+1\right)$$

Solution Steps

Calculate \sqrt{x-y^{2}} to the power of 2 and get x-y^{2}.

$$factor(\frac{x-y^{2}+\left(\sqrt{y+x^{2}}\right)^{2}}{\frac{x+y}{y-x}})$$

Calculate \sqrt{y+x^{2}} to the power of 2 and get y+x^{2}.

$$factor(\frac{x-y^{2}+y+x^{2}}{\frac{x+y}{y-x}})$$

Divide x-y^{2}+y+x^{2} by \frac{x+y}{y-x} by multiplying x-y^{2}+y+x^{2} by the reciprocal of \frac{x+y}{y-x}.

$$factor(\frac{\left(x-y^{2}+y+x^{2}\right)\left(y-x\right)}{x+y})$$

Factor the expressions that are not already factored in \frac{\left(x-y^{2}+y+x^{2}\right)\left(y-x\right)}{x+y}.

$$factor(\frac{\left(x+y\right)\left(-x+y\right)\left(x-y+1\right)}{x+y})$$

Cancel out x+y in both numerator and denominator.

$$factor(\left(-x+y\right)\left(x-y+1\right))$$

Expand the expression.

$$factor(-x^{2}+2xy-x-y^{2}+y)$$

Consider -x^{2}+2xy-x-y^{2}+y as a polynomial over variable x.

$$-x^{2}+\left(2y-1\right)x-y^{2}+y$$

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power -x^{2} and n divides the constant factor -y^{2}+y. One such factor is -x+y. Factor the polynomial by dividing it by this factor.

$$\left(-x+y\right)\left(x-y+1\right)$$