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

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

Evaluate

$$2x^{2}-3$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4+2\left(x-2\right)\left(x+4\right)-\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by x-2.

$$x^{2}-4x+4+\left(2x-4\right)\left(x+4\right)-\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2x-4 by x+4 and combine like terms.

$$x^{2}-4x+4+2x^{2}+4x-16-\left(x-3\right)\left(x+3\right)$$

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$3x^{2}-4x+4+4x-16-\left(x-3\right)\left(x+3\right)$$

Combine -4x and 4x to get 0.

$$3x^{2}+4-16-\left(x-3\right)\left(x+3\right)$$

Subtract 16 from 4 to get -12.

$$3x^{2}-12-\left(x-3\right)\left(x+3\right)$$

Consider \left(x-3\right)\left(x+3\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 3.

$$3x^{2}-12-\left(x^{2}-9\right)$$

To find the opposite of x^{2}-9, find the opposite of each term.

$$3x^{2}-12-x^{2}+9$$

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

$$2x^{2}-12+9$$

Add -12 and 9 to get -3.

$$2x^{2}-3$$

Expand

$$2x^{2}-3$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-4x+4+2\left(x-2\right)\left(x+4\right)-\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by x-2.

$$x^{2}-4x+4+\left(2x-4\right)\left(x+4\right)-\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 2x-4 by x+4 and combine like terms.

$$x^{2}-4x+4+2x^{2}+4x-16-\left(x-3\right)\left(x+3\right)$$

Combine x^{2} and 2x^{2} to get 3x^{2}.

$$3x^{2}-4x+4+4x-16-\left(x-3\right)\left(x+3\right)$$

Combine -4x and 4x to get 0.

$$3x^{2}+4-16-\left(x-3\right)\left(x+3\right)$$

Subtract 16 from 4 to get -12.

$$3x^{2}-12-\left(x-3\right)\left(x+3\right)$$

Consider \left(x-3\right)\left(x+3\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 3.

$$3x^{2}-12-\left(x^{2}-9\right)$$

To find the opposite of x^{2}-9, find the opposite of each term.

$$3x^{2}-12-x^{2}+9$$

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

$$2x^{2}-12+9$$

Add -12 and 9 to get -3.

$$2x^{2}-3$$