Solve left(x-2right)^2+left(x-1right)^2+x^2+left(x+1right)^2+{left(x+2right)}^2

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

Combine -4x and -2x to get -6x.

$$2x^{2}-6x+4+1+x^{2}+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

Add 4 and 1 to get 5.

$$2x^{2}-6x+5+x^{2}+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

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

$$3x^{2}-6x+5+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

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

$$3x^{2}-6x+5+x^{2}+2x+1+\left(x+2\right)^{2}$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-6x+5+2x+1+\left(x+2\right)^{2}$$

Combine -6x and 2x to get -4x.

$$4x^{2}-4x+5+1+\left(x+2\right)^{2}$$

Add 5 and 1 to get 6.

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

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

$$4x^{2}-4x+6+x^{2}+4x+4$$

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}-4x+6+4x+4$$

Combine -4x and 4x to get 0.

$$5x^{2}+6+4$$

Add 6 and 4 to get 10.

$$5x^{2}+10$$

Expand

$$5x^{2}+10$$

Solution Steps

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

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

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

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

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

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

Combine -4x and -2x to get -6x.

$$2x^{2}-6x+4+1+x^{2}+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

Add 4 and 1 to get 5.

$$2x^{2}-6x+5+x^{2}+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

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

$$3x^{2}-6x+5+\left(x+1\right)^{2}+\left(x+2\right)^{2}$$

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

$$3x^{2}-6x+5+x^{2}+2x+1+\left(x+2\right)^{2}$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-6x+5+2x+1+\left(x+2\right)^{2}$$

Combine -6x and 2x to get -4x.

$$4x^{2}-4x+5+1+\left(x+2\right)^{2}$$

Add 5 and 1 to get 6.

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

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

$$4x^{2}-4x+6+x^{2}+4x+4$$

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}-4x+6+4x+4$$

Combine -4x and 4x to get 0.

$$5x^{2}+6+4$$

Add 6 and 4 to get 10.

$$5x^{2}+10$$