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

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

Evaluate

$$\left(x-2\right)^{3}+14x+7$$

Solution Steps

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

$$x^{3}-6x^{2}+12x-8-\left(x-3\right)^{2}+\left(x+4\right)^{2}$$

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

$$x^{3}-6x^{2}+12x-8-\left(x^{2}-6x+9\right)+\left(x+4\right)^{2}$$

To find the opposite of x^{2}-6x+9, find the opposite of each term.

$$x^{3}-6x^{2}+12x-8-x^{2}+6x-9+\left(x+4\right)^{2}$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$x^{3}-7x^{2}+12x-8+6x-9+\left(x+4\right)^{2}$$

Combine 12x and 6x to get 18x.

$$x^{3}-7x^{2}+18x-8-9+\left(x+4\right)^{2}$$

Subtract 9 from -8 to get -17.

$$x^{3}-7x^{2}+18x-17+\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{3}-7x^{2}+18x-17+x^{2}+8x+16$$

Combine -7x^{2} and x^{2} to get -6x^{2}.

$$x^{3}-6x^{2}+18x-17+8x+16$$

Combine 18x and 8x to get 26x.

$$x^{3}-6x^{2}+26x-17+16$$

Add -17 and 16 to get -1.

$$x^{3}-6x^{2}+26x-1$$

Expand

$$x^{3}-6x^{2}+26x-1$$

Solution Steps

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

$$x^{3}-6x^{2}+12x-8-\left(x-3\right)^{2}+\left(x+4\right)^{2}$$

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

$$x^{3}-6x^{2}+12x-8-\left(x^{2}-6x+9\right)+\left(x+4\right)^{2}$$

To find the opposite of x^{2}-6x+9, find the opposite of each term.

$$x^{3}-6x^{2}+12x-8-x^{2}+6x-9+\left(x+4\right)^{2}$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$x^{3}-7x^{2}+12x-8+6x-9+\left(x+4\right)^{2}$$

Combine 12x and 6x to get 18x.

$$x^{3}-7x^{2}+18x-8-9+\left(x+4\right)^{2}$$

Subtract 9 from -8 to get -17.

$$x^{3}-7x^{2}+18x-17+\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{3}-7x^{2}+18x-17+x^{2}+8x+16$$

Combine -7x^{2} and x^{2} to get -6x^{2}.

$$x^{3}-6x^{2}+18x-17+8x+16$$

Combine 18x and 8x to get 26x.

$$x^{3}-6x^{2}+26x-17+16$$

Add -17 and 16 to get -1.

$$x^{3}-6x^{2}+26x-1$$