Solve (x-3)^2-5(3-x)^2+5/2(x-3)^2-(-3+x)^2=

$$( x - 3 ) ^ { 2 } - 5 ( 3 - x ) ^ { 2 } + \frac { 5 } { 2 } ( x - 3 ) ^ { 2 } - ( - 3 + x ) ^ { 2 } =$$

Evaluate

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

Solution Steps

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

$$x^{2}-6x+9-5\left(3-x\right)^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$x^{2}-6x+9-5\left(9-6x+x^{2}\right)+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Use the distributive property to multiply -5 by 9-6x+x^{2}.

$$x^{2}-6x+9-45+30x-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Subtract 45 from 9 to get -36.

$$x^{2}-6x-36+30x-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Combine -6x and 30x to get 24x.

$$x^{2}+24x-36-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$-4x^{2}+24x-36+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$-4x^{2}+24x-36+\frac{5}{2}\left(x^{2}-6x+9\right)-\left(-3+x\right)^{2}$$

Use the distributive property to multiply \frac{5}{2} by x^{2}-6x+9.

$$-4x^{2}+24x-36+\frac{5}{2}x^{2}-15x+\frac{45}{2}-\left(-3+x\right)^{2}$$

Combine -4x^{2} and \frac{5}{2}x^{2} to get -\frac{3}{2}x^{2}.

$$-\frac{3}{2}x^{2}+24x-36-15x+\frac{45}{2}-\left(-3+x\right)^{2}$$

Combine 24x and -15x to get 9x.

$$-\frac{3}{2}x^{2}+9x-36+\frac{45}{2}-\left(-3+x\right)^{2}$$

Add -36 and \frac{45}{2} to get -\frac{27}{2}.

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-\left(-3+x\right)^{2}$$

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

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-\left(9-6x+x^{2}\right)$$

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

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-9+6x-x^{2}$$

Subtract 9 from -\frac{27}{2} to get -\frac{45}{2}.

$$-\frac{3}{2}x^{2}+9x-\frac{45}{2}+6x-x^{2}$$

Combine 9x and 6x to get 15x.

$$-\frac{3}{2}x^{2}+15x-\frac{45}{2}-x^{2}$$

Combine -\frac{3}{2}x^{2} and -x^{2} to get -\frac{5}{2}x^{2}.

$$-\frac{5}{2}x^{2}+15x-\frac{45}{2}$$

Expand

$$-\frac{5x^{2}}{2}+15x-\frac{45}{2}$$

Solution Steps

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

$$x^{2}-6x+9-5\left(3-x\right)^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$x^{2}-6x+9-5\left(9-6x+x^{2}\right)+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Use the distributive property to multiply -5 by 9-6x+x^{2}.

$$x^{2}-6x+9-45+30x-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Subtract 45 from 9 to get -36.

$$x^{2}-6x-36+30x-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

Combine -6x and 30x to get 24x.

$$x^{2}+24x-36-5x^{2}+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$-4x^{2}+24x-36+\frac{5}{2}\left(x-3\right)^{2}-\left(-3+x\right)^{2}$$

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

$$-4x^{2}+24x-36+\frac{5}{2}\left(x^{2}-6x+9\right)-\left(-3+x\right)^{2}$$

Use the distributive property to multiply \frac{5}{2} by x^{2}-6x+9.

$$-4x^{2}+24x-36+\frac{5}{2}x^{2}-15x+\frac{45}{2}-\left(-3+x\right)^{2}$$

Combine -4x^{2} and \frac{5}{2}x^{2} to get -\frac{3}{2}x^{2}.

$$-\frac{3}{2}x^{2}+24x-36-15x+\frac{45}{2}-\left(-3+x\right)^{2}$$

Combine 24x and -15x to get 9x.

$$-\frac{3}{2}x^{2}+9x-36+\frac{45}{2}-\left(-3+x\right)^{2}$$

Add -36 and \frac{45}{2} to get -\frac{27}{2}.

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-\left(-3+x\right)^{2}$$

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

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-\left(9-6x+x^{2}\right)$$

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

$$-\frac{3}{2}x^{2}+9x-\frac{27}{2}-9+6x-x^{2}$$

Subtract 9 from -\frac{27}{2} to get -\frac{45}{2}.

$$-\frac{3}{2}x^{2}+9x-\frac{45}{2}+6x-x^{2}$$

Combine 9x and 6x to get 15x.

$$-\frac{3}{2}x^{2}+15x-\frac{45}{2}-x^{2}$$

Combine -\frac{3}{2}x^{2} and -x^{2} to get -\frac{5}{2}x^{2}.

$$-\frac{5}{2}x^{2}+15x-\frac{45}{2}$$