Solve (x-3)^2+2(x-2)(x+7)-(x+2)(x-2)

$$( x - 3 ) ^ { 2 } + 2 ( x - 2 ) ( x + 7 ) - ( x + 2 ) ( x - 2 )$$

Evaluate

$$2x^{2}+4x-15$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+2\left(x-2\right)\left(x+7\right)-\left(x+2\right)\left(x-2\right)$$

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

$$x^{2}-6x+9+\left(2x-4\right)\left(x+7\right)-\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by x+7 and combine like terms.

$$x^{2}-6x+9+2x^{2}+10x-28-\left(x+2\right)\left(x-2\right)$$

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

$$3x^{2}-6x+9+10x-28-\left(x+2\right)\left(x-2\right)$$

Combine -6x and 10x to get 4x.

$$3x^{2}+4x+9-28-\left(x+2\right)\left(x-2\right)$$

Subtract 28 from 9 to get -19.

$$3x^{2}+4x-19-\left(x+2\right)\left(x-2\right)$$

Consider \left(x+2\right)\left(x-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.

$$3x^{2}+4x-19-\left(x^{2}-4\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$3x^{2}+4x-19-x^{2}+4$$

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

$$2x^{2}+4x-19+4$$

Add -19 and 4 to get -15.

$$2x^{2}+4x-15$$

Expand

$$2x^{2}+4x-15$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9+2\left(x-2\right)\left(x+7\right)-\left(x+2\right)\left(x-2\right)$$

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

$$x^{2}-6x+9+\left(2x-4\right)\left(x+7\right)-\left(x+2\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by x+7 and combine like terms.

$$x^{2}-6x+9+2x^{2}+10x-28-\left(x+2\right)\left(x-2\right)$$

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

$$3x^{2}-6x+9+10x-28-\left(x+2\right)\left(x-2\right)$$

Combine -6x and 10x to get 4x.

$$3x^{2}+4x+9-28-\left(x+2\right)\left(x-2\right)$$

Subtract 28 from 9 to get -19.

$$3x^{2}+4x-19-\left(x+2\right)\left(x-2\right)$$

Consider \left(x+2\right)\left(x-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.

$$3x^{2}+4x-19-\left(x^{2}-4\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$3x^{2}+4x-19-x^{2}+4$$

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

$$2x^{2}+4x-19+4$$

Add -19 and 4 to get -15.

$$2x^{2}+4x-15$$