Solve -4x^2-4y/(frac{a{3}+3)^2-(a/3-3)^2}

$$\frac { - 4 x ^ { 2 } - 4 y } { ( \frac { a } { 3 } + 3 ) ^ { 2 } - ( \frac { a } { 3 } - 3 ) ^ { 2 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{3}{3}.

$$\frac{-4x^{2}-4y}{\left(\frac{a}{3}+\frac{3\times 3}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

Since \frac{a}{3} and \frac{3\times 3}{3} have the same denominator, add them by adding their numerators.

$$\frac{-4x^{2}-4y}{\left(\frac{a+3\times 3}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

Do the multiplications in a+3\times 3.

$$\frac{-4x^{2}-4y}{\left(\frac{a+9}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

To raise \frac{a+9}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a}{3}-3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{3}{3}.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a}{3}-\frac{3\times 3}{3}\right)^{2}}$$

Since \frac{a}{3} and \frac{3\times 3}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a-3\times 3}{3}\right)^{2}}$$

Do the multiplications in a-3\times 3.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a-9}{3}\right)^{2}}$$

To raise \frac{a-9}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\frac{\left(a-9\right)^{2}}{3^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\frac{\left(a-9\right)^{2}}{9}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 3^{2}.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{9}-\frac{\left(a-9\right)^{2}}{9}}$$

Since \frac{\left(a+9\right)^{2}}{9} and \frac{\left(a-9\right)^{2}}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}-\left(a-9\right)^{2}}{9}}$$

Do the multiplications in \left(a+9\right)^{2}-\left(a-9\right)^{2}.

$$\frac{-4x^{2}-4y}{\frac{a^{2}+18a+81-a^{2}+18a-81}{9}}$$

Combine like terms in a^{2}+18a+81-a^{2}+18a-81.

$$\frac{-4x^{2}-4y}{\frac{36a}{9}}$$

Divide -4x^{2}-4y by \frac{36a}{9} by multiplying -4x^{2}-4y by the reciprocal of \frac{36a}{9}.

$$\frac{\left(-4x^{2}-4y\right)\times 9}{36a}$$

Cancel out 9 in both numerator and denominator.

$$\frac{-4x^{2}-4y}{4a}$$

Factor the expressions that are not already factored.

$$\frac{4\left(-x^{2}-y\right)}{4a}$$

Cancel out 4 in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{3}{3}.

$$\frac{-4x^{2}-4y}{\left(\frac{a}{3}+\frac{3\times 3}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

Since \frac{a}{3} and \frac{3\times 3}{3} have the same denominator, add them by adding their numerators.

$$\frac{-4x^{2}-4y}{\left(\frac{a+3\times 3}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

Do the multiplications in a+3\times 3.

$$\frac{-4x^{2}-4y}{\left(\frac{a+9}{3}\right)^{2}-\left(\frac{a}{3}-3\right)^{2}}$$

To raise \frac{a+9}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a}{3}-3\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{3}{3}.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a}{3}-\frac{3\times 3}{3}\right)^{2}}$$

Since \frac{a}{3} and \frac{3\times 3}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a-3\times 3}{3}\right)^{2}}$$

Do the multiplications in a-3\times 3.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\left(\frac{a-9}{3}\right)^{2}}$$

To raise \frac{a-9}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\frac{\left(a-9\right)^{2}}{3^{2}}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{3^{2}}-\frac{\left(a-9\right)^{2}}{9}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 3^{2}.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}}{9}-\frac{\left(a-9\right)^{2}}{9}}$$

Since \frac{\left(a+9\right)^{2}}{9} and \frac{\left(a-9\right)^{2}}{9} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}-4y}{\frac{\left(a+9\right)^{2}-\left(a-9\right)^{2}}{9}}$$

Do the multiplications in \left(a+9\right)^{2}-\left(a-9\right)^{2}.

$$\frac{-4x^{2}-4y}{\frac{a^{2}+18a+81-a^{2}+18a-81}{9}}$$

Combine like terms in a^{2}+18a+81-a^{2}+18a-81.

$$\frac{-4x^{2}-4y}{\frac{36a}{9}}$$

Divide -4x^{2}-4y by \frac{36a}{9} by multiplying -4x^{2}-4y by the reciprocal of \frac{36a}{9}.

$$\frac{\left(-4x^{2}-4y\right)\times 9}{36a}$$

Cancel out 9 in both numerator and denominator.

$$\frac{-4x^{2}-4y}{4a}$$

Factor the expressions that are not already factored.

$$\frac{4\left(-x^{2}-y\right)}{4a}$$

Cancel out 4 in both numerator and denominator.

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