Solve (x-1)^2+(x+2)^2-(x-3)(x+3)-22

$$( x - 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } - ( x - 3 ) ( x + 3 ) - 22$$

Evaluate

$$\left(x-2\right)\left(x+4\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)-22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)-22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)-22$$

Combine -2x and 4x to get 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)-22$$

Add 1 and 4 to get 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)-22$$

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.

$$2x^{2}+2x+5-\left(x^{2}-9\right)-22$$

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

$$2x^{2}+2x+5-x^{2}+9-22$$

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

$$x^{2}+2x+5+9-22$$

Add 5 and 9 to get 14.

$$x^{2}+2x+14-22$$

Subtract 22 from 14 to get -8.

$$x^{2}+2x-8$$

Expand

$$x^{2}+2x-8$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1+\left(x+2\right)^{2}-\left(x-3\right)\left(x+3\right)-22$$

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

$$x^{2}-2x+1+x^{2}+4x+4-\left(x-3\right)\left(x+3\right)-22$$

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

$$2x^{2}-2x+1+4x+4-\left(x-3\right)\left(x+3\right)-22$$

Combine -2x and 4x to get 2x.

$$2x^{2}+2x+1+4-\left(x-3\right)\left(x+3\right)-22$$

Add 1 and 4 to get 5.

$$2x^{2}+2x+5-\left(x-3\right)\left(x+3\right)-22$$

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.

$$2x^{2}+2x+5-\left(x^{2}-9\right)-22$$

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

$$2x^{2}+2x+5-x^{2}+9-22$$

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

$$x^{2}+2x+5+9-22$$

Add 5 and 9 to get 14.

$$x^{2}+2x+14-22$$

Subtract 22 from 14 to get -8.

$$x^{2}+2x-8$$