Solve 17:2(a-3)(a+2)-(4+a)(4-a)

$$17 : 2 ( a - 3 ) ( a + 2 ) - ( 4 + a ) ( 4 - a )$$

Evaluate

$$\frac{19a^{2}}{2}-\frac{17a}{2}-67$$

Solution Steps

Use the distributive property to multiply \frac{17}{2} by a-3.

$$\left(\frac{17}{2}a+\frac{17}{2}\left(-3\right)\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Express \frac{17}{2}\left(-3\right) as a single fraction.

$$\left(\frac{17}{2}a+\frac{17\left(-3\right)}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Multiply 17 and -3 to get -51.

$$\left(\frac{17}{2}a+\frac{-51}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Fraction \frac{-51}{2} can be rewritten as -\frac{51}{2} by extracting the negative sign.

$$\left(\frac{17}{2}a-\frac{51}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Apply the distributive property by multiplying each term of \frac{17}{2}a-\frac{51}{2} by each term of a+2.

$$\frac{17}{2}aa+\frac{17}{2}a\times 2-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Multiply a and a to get a^{2}.

$$\frac{17}{2}a^{2}+\frac{17}{2}a\times 2-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Cancel out 2 and 2.

$$\frac{17}{2}a^{2}+17a-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Combine 17a and -\frac{51}{2}a to get -\frac{17}{2}a.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Cancel out 2 and 2.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(4+a\right)\left(4-a\right)$$

Consider \left(4+a\right)\left(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}.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(4^{2}-a^{2}\right)$$

Calculate 4 to the power of 2 and get 16.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(16-a^{2}\right)$$

To find the opposite of 16-a^{2}, find the opposite of each term.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-16-\left(-a^{2}\right)$$

The opposite of -a^{2} is a^{2}.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-16+a^{2}$$

Subtract 16 from -51 to get -67.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-67+a^{2}$$

Combine \frac{17}{2}a^{2} and a^{2} to get \frac{19}{2}a^{2}.

$$\frac{19}{2}a^{2}-\frac{17}{2}a-67$$

Expand

$$\frac{19a^{2}}{2}-\frac{17a}{2}-67$$

Solution Steps

Use the distributive property to multiply \frac{17}{2} by a-3.

$$\left(\frac{17}{2}a+\frac{17}{2}\left(-3\right)\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Express \frac{17}{2}\left(-3\right) as a single fraction.

$$\left(\frac{17}{2}a+\frac{17\left(-3\right)}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Multiply 17 and -3 to get -51.

$$\left(\frac{17}{2}a+\frac{-51}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Fraction \frac{-51}{2} can be rewritten as -\frac{51}{2} by extracting the negative sign.

$$\left(\frac{17}{2}a-\frac{51}{2}\right)\left(a+2\right)-\left(4+a\right)\left(4-a\right)$$

Apply the distributive property by multiplying each term of \frac{17}{2}a-\frac{51}{2} by each term of a+2.

$$\frac{17}{2}aa+\frac{17}{2}a\times 2-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Multiply a and a to get a^{2}.

$$\frac{17}{2}a^{2}+\frac{17}{2}a\times 2-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Cancel out 2 and 2.

$$\frac{17}{2}a^{2}+17a-\frac{51}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Combine 17a and -\frac{51}{2}a to get -\frac{17}{2}a.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-\frac{51}{2}\times 2-\left(4+a\right)\left(4-a\right)$$

Cancel out 2 and 2.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(4+a\right)\left(4-a\right)$$

Consider \left(4+a\right)\left(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}.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(4^{2}-a^{2}\right)$$

Calculate 4 to the power of 2 and get 16.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-\left(16-a^{2}\right)$$

To find the opposite of 16-a^{2}, find the opposite of each term.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-16-\left(-a^{2}\right)$$

The opposite of -a^{2} is a^{2}.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-51-16+a^{2}$$

Subtract 16 from -51 to get -67.

$$\frac{17}{2}a^{2}-\frac{17}{2}a-67+a^{2}$$

Combine \frac{17}{2}a^{2} and a^{2} to get \frac{19}{2}a^{2}.

$$\frac{19}{2}a^{2}-\frac{17}{2}a-67$$