Solve 2(x-2)^2-7x-(x-3)^2-1

$$2 ( x - 2 ) ^ { 2 } - 7 x - ( x - 3 ) ^ { 2 } - 1$$

Evaluate

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

Solution Steps

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

$$2\left(x^{2}-4x+4\right)-7x-\left(x-3\right)^{2}-1$$

Use the distributive property to multiply 2 by x^{2}-4x+4.

$$2x^{2}-8x+8-7x-\left(x-3\right)^{2}-1$$

Combine -8x and -7x to get -15x.

$$2x^{2}-15x+8-\left(x-3\right)^{2}-1$$

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

$$2x^{2}-15x+8-\left(x^{2}-6x+9\right)-1$$

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

$$2x^{2}-15x+8-x^{2}+6x-9-1$$

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

$$x^{2}-15x+8+6x-9-1$$

Combine -15x and 6x to get -9x.

$$x^{2}-9x+8-9-1$$

Subtract 9 from 8 to get -1.

$$x^{2}-9x-1-1$$

Subtract 1 from -1 to get -2.

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

Expand

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

Solution Steps

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

$$2\left(x^{2}-4x+4\right)-7x-\left(x-3\right)^{2}-1$$

Use the distributive property to multiply 2 by x^{2}-4x+4.

$$2x^{2}-8x+8-7x-\left(x-3\right)^{2}-1$$

Combine -8x and -7x to get -15x.

$$2x^{2}-15x+8-\left(x-3\right)^{2}-1$$

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

$$2x^{2}-15x+8-\left(x^{2}-6x+9\right)-1$$

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

$$2x^{2}-15x+8-x^{2}+6x-9-1$$

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

$$x^{2}-15x+8+6x-9-1$$

Combine -15x and 6x to get -9x.

$$x^{2}-9x+8-9-1$$

Subtract 9 from 8 to get -1.

$$x^{2}-9x-1-1$$

Subtract 1 from -1 to get -2.

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