Solve (7/2-1/4a)(7/2+1/4a)+a^2(a-2)(a+2)-(2a-frac{1}{2})(frac{a-2}{2})(2a+frac{1}{2})(frac{a}{2}+1)

$$( \frac { 7 } { 2 } - \frac { 1 } { 4 } a ) ( \frac { 7 } { 2 } + \frac { 1 } { 4 } a ) + a ^ { 2 } ( a - 2 ) ( a + 2 ) - ( 2 a - \frac { 1 } { 2 } ) ( \frac { a - 2 } { 2 } ) ( 2 a + \frac { 1 } { 2 } ) ( \frac { a } { 2 } + 1 )$$

Evaluate

$$12$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$\left(\frac{7}{2}-\frac{1}{4}a\right)\left(\frac{7}{2}+\frac{1}{4}a\right)+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{a-2}{2}\left(2a+\frac{1}{2}\right)\left(\frac{a}{2}+\frac{2}{2}\right)$$

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

$$\left(\frac{7}{2}-\frac{1}{4}a\right)\left(\frac{7}{2}+\frac{1}{4}a\right)+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{a-2}{2}\left(2a+\frac{1}{2}\right)\times \frac{a+2}{2}$$

Multiply \frac{a-2}{2} times \frac{a+2}{2} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{7}{2}-\frac{1}{4}a\right)\left(\frac{7}{2}+\frac{1}{4}a\right)+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Consider \left(\frac{7}{2}-\frac{1}{4}a\right)\left(\frac{7}{2}+\frac{1}{4}a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{7}{2}.

$$\frac{49}{4}-\left(\frac{1}{4}a\right)^{2}+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Expand \left(\frac{1}{4}a\right)^{2}.

$$\frac{49}{4}-\left(\frac{1}{4}\right)^{2}a^{2}+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Calculate \frac{1}{4} to the power of 2 and get \frac{1}{16}.

$$\frac{49}{4}-\frac{1}{16}a^{2}+a^{2}\left(a-2\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

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

$$\frac{49}{4}-\frac{1}{16}a^{2}+\left(a^{3}-2a^{2}\right)\left(a+2\right)-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Use the distributive property to multiply a^{3}-2a^{2} by a+2 and combine like terms.

$$\frac{49}{4}-\frac{1}{16}a^{2}+a^{4}-4a^{2}-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Combine -\frac{1}{16}a^{2} and -4a^{2} to get -\frac{65}{16}a^{2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{2\times 2}\left(2a+\frac{1}{2}\right)$$

Multiply 2 and 2 to get 4.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(2a-\frac{1}{2}\right)\times \frac{\left(a-2\right)\left(a+2\right)}{4}\left(2a+\frac{1}{2}\right)$$

Use the distributive property to multiply 2a-\frac{1}{2} by \frac{\left(a-2\right)\left(a+2\right)}{4}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(2a\times \frac{\left(a-2\right)\left(a+2\right)}{4}-\frac{1}{2}\times \frac{\left(a-2\right)\left(a+2\right)}{4}\right)\left(2a+\frac{1}{2}\right)$$

Consider \left(a-2\right)\left(a+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(2a\times \frac{a^{2}-4}{4}-\frac{1}{2}\times \frac{\left(a-2\right)\left(a+2\right)}{4}\right)\left(2a+\frac{1}{2}\right)$$

Cancel out 4, the greatest common factor in 2 and 4.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{a^{2}-4}{2}a-\frac{1}{2}\times \frac{\left(a-2\right)\left(a+2\right)}{4}\right)\left(2a+\frac{1}{2}\right)$$

Consider \left(a-2\right)\left(a+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{a^{2}-4}{2}a-\frac{1}{2}\times \frac{a^{2}-4}{4}\right)\left(2a+\frac{1}{2}\right)$$

Multiply -\frac{1}{2} times \frac{a^{2}-4}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{a^{2}-4}{2}a+\frac{-\left(a^{2}-4\right)}{2\times 4}\right)\left(2a+\frac{1}{2}\right)$$

Use the distributive property to multiply \frac{a^{2}-4}{2}a+\frac{-\left(a^{2}-4\right)}{2\times 4} by 2a+\frac{1}{2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(2\times \frac{a^{2}-4}{2}a^{2}+\frac{1}{2}\times \frac{a^{2}-4}{2}a+2\times \frac{-\left(a^{2}-4\right)}{2\times 4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Cancel out 2 and 2.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\left(a^{2}-4\right)a^{2}+\frac{1}{2}\times \frac{a^{2}-4}{2}a+2\times \frac{-\left(a^{2}-4\right)}{2\times 4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Use the distributive property to multiply a^{2}-4 by a^{2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{1}{2}\times \frac{a^{2}-4}{2}a+2\times \frac{-\left(a^{2}-4\right)}{2\times 4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Multiply \frac{1}{2} times \frac{a^{2}-4}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{a^{2}-4}{2\times 2}a+2\times \frac{-\left(a^{2}-4\right)}{2\times 4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Express \frac{a^{2}-4}{2\times 2}a as a single fraction.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+2\times \frac{-\left(a^{2}-4\right)}{2\times 4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Multiply 2 and 4 to get 8.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+2\times \frac{-\left(a^{2}-4\right)}{8}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Cancel out 8, the greatest common factor in 2 and 8.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)}{4}a+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Express \frac{-\left(a^{2}-4\right)}{4}a as a single fraction.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{2\times 4}\right)$$

Multiply 2 and 4 to get 8.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{1}{2}\times \frac{-\left(a^{2}-4\right)}{8}\right)$$

Multiply \frac{1}{2} times \frac{-\left(a^{2}-4\right)}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{4}-4a^{2} times \frac{2\times 2}{2\times 2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{\left(a^{4}-4a^{2}\right)\times 2\times 2}{2\times 2}+\frac{\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Since \frac{\left(a^{4}-4a^{2}\right)\times 2\times 2}{2\times 2} and \frac{\left(a^{2}-4\right)a}{2\times 2} have the same denominator, add them by adding their numerators.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{\left(a^{4}-4a^{2}\right)\times 2\times 2+\left(a^{2}-4\right)a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Do the multiplications in \left(a^{4}-4a^{2}\right)\times 2\times 2+\left(a^{2}-4\right)a.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{4a^{4}-16a^{2}+a^{3}-4a}{2\times 2}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2\times 2.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{4a^{4}-16a^{2}+a^{3}-4a}{4}+\frac{-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Since \frac{4a^{4}-16a^{2}+a^{3}-4a}{4} and \frac{-\left(a^{2}-4\right)a}{4} have the same denominator, add them by adding their numerators.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{4a^{4}-16a^{2}+a^{3}-4a-\left(a^{2}-4\right)a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Do the multiplications in 4a^{4}-16a^{2}+a^{3}-4a-\left(a^{2}-4\right)a.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{4a^{4}-16a^{2}+a^{3}-4a-a^{3}+4a}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Combine like terms in 4a^{4}-16a^{2}+a^{3}-4a-a^{3}+4a.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{4a^{4}-16a^{2}}{4}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Divide each term of 4a^{4}-16a^{2} by 4 to get a^{4}-4a^{2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(a^{4}-4a^{2}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{4}-4a^{2} times \frac{2\times 8}{2\times 8}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\left(\frac{\left(a^{4}-4a^{2}\right)\times 2\times 8}{2\times 8}+\frac{-\left(a^{2}-4\right)}{2\times 8}\right)$$

Since \frac{\left(a^{4}-4a^{2}\right)\times 2\times 8}{2\times 8} and \frac{-\left(a^{2}-4\right)}{2\times 8} have the same denominator, add them by adding their numerators.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\frac{\left(a^{4}-4a^{2}\right)\times 2\times 8-\left(a^{2}-4\right)}{2\times 8}$$

Do the multiplications in \left(a^{4}-4a^{2}\right)\times 2\times 8-\left(a^{2}-4\right).

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\frac{16a^{4}-64a^{2}-a^{2}+4}{2\times 8}$$

Combine like terms in 16a^{4}-64a^{2}-a^{2}+4.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\frac{16a^{4}-65a^{2}+4}{2\times 8}$$

Multiply 2 and 8 to get 16.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}-\frac{16a^{4}-65a^{2}+4}{16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 16 is 16. Multiply \frac{49}{4} times \frac{4}{4}.

$$\frac{49\times 4}{16}-\frac{65}{16}a^{2}+a^{4}-\frac{16a^{4}-65a^{2}+4}{16}$$

Since \frac{49\times 4}{16} and \frac{16a^{4}-65a^{2}+4}{16} have the same denominator, subtract them by subtracting their numerators.

$$\frac{49\times 4-\left(16a^{4}-65a^{2}+4\right)}{16}-\frac{65}{16}a^{2}+a^{4}$$

Do the multiplications in 49\times 4-\left(16a^{4}-65a^{2}+4\right).

$$\frac{196-16a^{4}+65a^{2}-4}{16}-\frac{65}{16}a^{2}+a^{4}$$

Combine like terms in 196-16a^{4}+65a^{2}-4.

$$\frac{192-16a^{4}+65a^{2}}{16}-\frac{65}{16}a^{2}+a^{4}$$

Divide each term of 192-16a^{4}+65a^{2} by 16 to get 12+\frac{65}{16}a^{2}-a^{4}.

$$12+\frac{65}{16}a^{2}-a^{4}-\frac{65}{16}a^{2}+a^{4}$$

Combine \frac{65}{16}a^{2} and -\frac{65}{16}a^{2} to get 0.

$$12-a^{4}+a^{4}$$

Combine -a^{4} and a^{4} to get 0.

$$12$$

Factor

$$2^{2}\times 3$$

Solution Steps

Factor out \frac{1}{16}.

$$\frac{\left(14-a\right)\left(14+a\right)+16a^{2}\left(a-2\right)\left(a+2\right)-\left(4a-1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16}$$

Simplify.

$$12$$