Solve (x-1/2)(x+2)+(x-1/3)(x+3)-x(25/6+x)

$$( x - \frac { 1 } { 2 } ) ( x + 2 ) + ( x - \frac { 1 } { 3 } ) ( x + 3 ) - x ( \frac { 25 } { 6 } + x )$$

Evaluate

$$x^{2}-2$$

Solution Steps

Apply the distributive property by multiplying each term of x-\frac{1}{2} by each term of x+2.

$$x^{2}+2x-\frac{1}{2}x-\frac{1}{2}\times 2+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Combine 2x and -\frac{1}{2}x to get \frac{3}{2}x.

$$x^{2}+\frac{3}{2}x-\frac{1}{2}\times 2+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Cancel out 2 and 2.

$$x^{2}+\frac{3}{2}x-1+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Apply the distributive property by multiplying each term of x-\frac{1}{3} by each term of x+3.

$$x^{2}+\frac{3}{2}x-1+x^{2}+3x-\frac{1}{3}x-\frac{1}{3}\times 3-x\left(\frac{25}{6}+x\right)$$

Combine 3x and -\frac{1}{3}x to get \frac{8}{3}x.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-\frac{1}{3}\times 3-x\left(\frac{25}{6}+x\right)$$

Cancel out 3 and 3.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-1-x\left(\frac{25}{6}+x\right)$$

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

$$2x^{2}+\frac{3}{2}x-1+\frac{8}{3}x-1-x\left(\frac{25}{6}+x\right)$$

Combine \frac{3}{2}x and \frac{8}{3}x to get \frac{25}{6}x.

$$2x^{2}+\frac{25}{6}x-1-1-x\left(\frac{25}{6}+x\right)$$

Subtract 1 from -1 to get -2.

$$2x^{2}+\frac{25}{6}x-2-x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x by \frac{25}{6}+x.

$$2x^{2}+\frac{25}{6}x-2-\left(x\times \frac{25}{6}+x^{2}\right)$$

To find the opposite of x\times \frac{25}{6}+x^{2}, find the opposite of each term.

$$2x^{2}+\frac{25}{6}x-2-x\times \frac{25}{6}-x^{2}$$

Multiply -1 and \frac{25}{6} to get -\frac{25}{6}.

$$2x^{2}+\frac{25}{6}x-2-\frac{25}{6}x-x^{2}$$

Combine \frac{25}{6}x and -\frac{25}{6}x to get 0.

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

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

$$x^{2}-2$$

Expand

$$x^{2}-2$$

Solution Steps

Apply the distributive property by multiplying each term of x-\frac{1}{2} by each term of x+2.

$$x^{2}+2x-\frac{1}{2}x-\frac{1}{2}\times 2+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Combine 2x and -\frac{1}{2}x to get \frac{3}{2}x.

$$x^{2}+\frac{3}{2}x-\frac{1}{2}\times 2+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Cancel out 2 and 2.

$$x^{2}+\frac{3}{2}x-1+\left(x-\frac{1}{3}\right)\left(x+3\right)-x\left(\frac{25}{6}+x\right)$$

Apply the distributive property by multiplying each term of x-\frac{1}{3} by each term of x+3.

$$x^{2}+\frac{3}{2}x-1+x^{2}+3x-\frac{1}{3}x-\frac{1}{3}\times 3-x\left(\frac{25}{6}+x\right)$$

Combine 3x and -\frac{1}{3}x to get \frac{8}{3}x.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-\frac{1}{3}\times 3-x\left(\frac{25}{6}+x\right)$$

Cancel out 3 and 3.

$$x^{2}+\frac{3}{2}x-1+x^{2}+\frac{8}{3}x-1-x\left(\frac{25}{6}+x\right)$$

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

$$2x^{2}+\frac{3}{2}x-1+\frac{8}{3}x-1-x\left(\frac{25}{6}+x\right)$$

Combine \frac{3}{2}x and \frac{8}{3}x to get \frac{25}{6}x.

$$2x^{2}+\frac{25}{6}x-1-1-x\left(\frac{25}{6}+x\right)$$

Subtract 1 from -1 to get -2.

$$2x^{2}+\frac{25}{6}x-2-x\left(\frac{25}{6}+x\right)$$

Use the distributive property to multiply x by \frac{25}{6}+x.

$$2x^{2}+\frac{25}{6}x-2-\left(x\times \frac{25}{6}+x^{2}\right)$$

To find the opposite of x\times \frac{25}{6}+x^{2}, find the opposite of each term.

$$2x^{2}+\frac{25}{6}x-2-x\times \frac{25}{6}-x^{2}$$

Multiply -1 and \frac{25}{6} to get -\frac{25}{6}.

$$2x^{2}+\frac{25}{6}x-2-\frac{25}{6}x-x^{2}$$

Combine \frac{25}{6}x and -\frac{25}{6}x to get 0.

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

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

$$x^{2}-2$$