Solve 2(a+3)^2-4(a+3)(a-3)+3(a-2)^2

$$2 ( a + 3 ) ^ { 2 } - 4 ( a + 3 ) ( a - 3 ) + 3 ( a - 2 ) ^ { 2 }$$

Evaluate

$$a^{2}+66$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3\right)^{2}.

$$2\left(a^{2}+6a+9\right)-4\left(a+3\right)\left(a-3\right)+3\left(a-2\right)^{2}$$

Use the distributive property to multiply 2 by a^{2}+6a+9.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3\left(a-2\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-2\right)^{2}.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3\left(a^{2}-4a+4\right)$$

Use the distributive property to multiply 3 by a^{2}-4a+4.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3a^{2}-12a+12$$

Use the distributive property to multiply -4 by a+3.

$$2a^{2}+12a+18+\left(-4a-12\right)\left(a-3\right)+3a^{2}-12a+12$$

Use the distributive property to multiply -4a-12 by a-3 and combine like terms.

$$2a^{2}+12a+18-4a^{2}+36+3a^{2}-12a+12$$

Combine 2a^{2} and -4a^{2} to get -2a^{2}.

$$-2a^{2}+12a+18+36+3a^{2}-12a+12$$

Add 18 and 36 to get 54.

$$-2a^{2}+12a+54+3a^{2}-12a+12$$

Combine -2a^{2} and 3a^{2} to get a^{2}.

$$a^{2}+12a+54-12a+12$$

Combine 12a and -12a to get 0.

$$a^{2}+54+12$$

Add 54 and 12 to get 66.

$$a^{2}+66$$

Expand

$$a^{2}+66$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3\right)^{2}.

$$2\left(a^{2}+6a+9\right)-4\left(a+3\right)\left(a-3\right)+3\left(a-2\right)^{2}$$

Use the distributive property to multiply 2 by a^{2}+6a+9.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3\left(a-2\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-2\right)^{2}.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3\left(a^{2}-4a+4\right)$$

Use the distributive property to multiply 3 by a^{2}-4a+4.

$$2a^{2}+12a+18-4\left(a+3\right)\left(a-3\right)+3a^{2}-12a+12$$

Use the distributive property to multiply -4 by a+3.

$$2a^{2}+12a+18+\left(-4a-12\right)\left(a-3\right)+3a^{2}-12a+12$$

Use the distributive property to multiply -4a-12 by a-3 and combine like terms.

$$2a^{2}+12a+18-4a^{2}+36+3a^{2}-12a+12$$

Combine 2a^{2} and -4a^{2} to get -2a^{2}.

$$-2a^{2}+12a+18+36+3a^{2}-12a+12$$

Add 18 and 36 to get 54.

$$-2a^{2}+12a+54+3a^{2}-12a+12$$

Combine -2a^{2} and 3a^{2} to get a^{2}.

$$a^{2}+12a+54-12a+12$$

Combine 12a and -12a to get 0.

$$a^{2}+54+12$$

Add 54 and 12 to get 66.

$$a^{2}+66$$