Solve (a+z)(a-z)^2/2a*z+2a/a^2+z^2-2az

$$\frac { ( a + z ) ( a - z ) ^ { 2 } } { 2 a } \cdot \frac { z + 2 a } { a ^ { 2 } + z ^ { 2 } - 2 a z }$$

Evaluate

$$\frac{\left(z+a\right)\left(z+2a\right)}{2a}$$

Short Solution Steps

Use the distributive property to multiply a+z by \left(a-z\right)^{2}.

$$\frac{a\left(a-z\right)^{2}+z\left(a-z\right)^{2}}{2a}\times \frac{z+2a}{a^{2}+z^{2}-2az}$$

Multiply \frac{a\left(a-z\right)^{2}+z\left(a-z\right)^{2}}{2a} times \frac{z+2a}{a^{2}+z^{2}-2az} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a\left(a-z\right)^{2}+z\left(a-z\right)^{2}\right)\left(z+2a\right)}{2a\left(a^{2}+z^{2}-2az\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(z+a\right)\left(z+2a\right)\left(-z+a\right)^{2}}{2a\left(-z+a\right)^{2}}$$

Cancel out \left(-z+a\right)^{2} in both numerator and denominator.

$$\frac{\left(z+a\right)\left(z+2a\right)}{2a}$$

Expand the expression.

$$\frac{z^{2}+3az+2a^{2}}{2a}$$

Expand

$$\frac{z^{2}}{2a}+\frac{3z}{2}+a$$

Short Solution Steps

Use the distributive property to multiply a+z by \left(a-z\right)^{2}.

$$\frac{a\left(a-z\right)^{2}+z\left(a-z\right)^{2}}{2a}\times \frac{z+2a}{a^{2}+z^{2}-2az}$$

Multiply \frac{a\left(a-z\right)^{2}+z\left(a-z\right)^{2}}{2a} times \frac{z+2a}{a^{2}+z^{2}-2az} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a\left(a-z\right)^{2}+z\left(a-z\right)^{2}\right)\left(z+2a\right)}{2a\left(a^{2}+z^{2}-2az\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(z+a\right)\left(z+2a\right)\left(-z+a\right)^{2}}{2a\left(-z+a\right)^{2}}$$

Cancel out \left(-z+a\right)^{2} in both numerator and denominator.

$$\frac{\left(z+a\right)\left(z+2a\right)}{2a}$$

Expand the expression.

$$\frac{z^{2}+3az+2a^{2}}{2a}$$