Solve -1/2a-a^2/2+3a/2/sqrt{1+(frac{1{2})^2}}

$$\frac { - \frac { 1 } { 2 } a - \frac { a ^ { 2 } } { 2 } + \frac { 3 a } { 2 } } { \sqrt { 1 + ( \frac { 1 } { 2 } ) ^ { 2 } } }$$

Evaluate

$$\frac{\sqrt{5}a\left(2-a\right)}{5}$$

Solution Steps

Since -\frac{a^{2}}{2} and \frac{3a}{2} have the same denominator, add them by adding their numerators.

$$\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{1+\left(\frac{1}{2}\right)^{2}}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{1+\frac{1}{4}}}$$

Add 1 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{\frac{5}{4}}}$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\frac{\sqrt{5}}{\sqrt{4}}}$$

Calculate the square root of 4 and get 2.

$$\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\frac{\sqrt{5}}{2}}$$

Divide -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by \frac{\sqrt{5}}{2} by multiplying -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by the reciprocal of \frac{\sqrt{5}}{2}.

$$\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2}{\sqrt{5}}$$

Rationalize the denominator of \frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2\sqrt{5}}{5}$$

Use the distributive property to multiply -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by 2.

$$\frac{\left(-a+2\times \frac{-a^{2}+3a}{2}\right)\sqrt{5}}{5}$$

Cancel out 2 and 2.

$$\frac{\left(-a-a^{2}+3a\right)\sqrt{5}}{5}$$

Combine -a and 3a to get 2a.

$$\frac{\left(2a-a^{2}\right)\sqrt{5}}{5}$$

Use the distributive property to multiply 2a-a^{2} by \sqrt{5}.

$$\frac{2a\sqrt{5}-a^{2}\sqrt{5}}{5}$$

Factor

$$\frac{\sqrt{5}a\left(2-a\right)}{5}$$

Solution Steps

Since -\frac{a^{2}}{2} and \frac{3a}{2} have the same denominator, add them by adding their numerators.

$$factor(\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{1+\left(\frac{1}{2}\right)^{2}}})$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$factor(\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{1+\frac{1}{4}}})$$

Add 1 and \frac{1}{4} to get \frac{5}{4}.

$$factor(\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\sqrt{\frac{5}{4}}})$$

Rewrite the square root of the division \sqrt{\frac{5}{4}} as the division of square roots \frac{\sqrt{5}}{\sqrt{4}}.

$$factor(\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\frac{\sqrt{5}}{\sqrt{4}}})$$

Calculate the square root of 4 and get 2.

$$factor(\frac{-\frac{1}{2}a+\frac{-a^{2}+3a}{2}}{\frac{\sqrt{5}}{2}})$$

Divide -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by \frac{\sqrt{5}}{2} by multiplying -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by the reciprocal of \frac{\sqrt{5}}{2}.

$$factor(\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2}{\sqrt{5}})$$

Rationalize the denominator of \frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$factor(\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2\sqrt{5}}{\left(\sqrt{5}\right)^{2}})$$

The square of \sqrt{5} is 5.

$$factor(\frac{\left(-\frac{1}{2}a+\frac{-a^{2}+3a}{2}\right)\times 2\sqrt{5}}{5})$$

Use the distributive property to multiply -\frac{1}{2}a+\frac{-a^{2}+3a}{2} by 2.

$$factor(\frac{\left(-a+2\times \frac{-a^{2}+3a}{2}\right)\sqrt{5}}{5})$$

Cancel out 2 and 2.

$$factor(\frac{\left(-a-a^{2}+3a\right)\sqrt{5}}{5})$$

Combine -a and 3a to get 2a.

$$factor(\frac{\left(2a-a^{2}\right)\sqrt{5}}{5})$$

Use the distributive property to multiply 2a-a^{2} by \sqrt{5}.

$$factor(\frac{2a\sqrt{5}-a^{2}\sqrt{5}}{5})$$

Consider 2a\sqrt{5}-a^{2}\sqrt{5}. Factor out a\sqrt{5}.

$$a\sqrt{5}\left(2-a\right)$$

Rewrite the complete factored expression.

$$\frac{a\left(-a+2\right)\sqrt{5}}{5}$$