Solve 1/2x(5x-1)+5x+2=1/2(x+1)(5(x+1)-1)

$$\frac{ 1 }{ 2 } x(5x-1)+5x+2= \frac{ 1 }{ 2 } (x+1)(5(x+1)-1)$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}x by 5x-1.

$$\frac{1}{2}x\times 5x+\frac{1}{2}x\left(-1\right)+5x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

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

$$\frac{1}{2}x^{2}\times 5+\frac{1}{2}x\left(-1\right)+5x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Combine -\frac{1}{2}x and 5x to get \frac{9}{2}x.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

Use the distributive property to multiply 5 by x+1.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5x+5-1\right)$$

Subtract 1 from 5 to get 4.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5x+4\right)$$

Use the distributive property to multiply \frac{1}{2} by x+1.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\left(\frac{1}{2}x+\frac{1}{2}\right)\left(5x+4\right)$$

Apply the distributive property by multiplying each term of \frac{1}{2}x+\frac{1}{2} by each term of 5x+4.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}x\times 5x+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

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

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}x^{2}\times 5+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{4}{2}x+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Divide 4 by 2 to get 2.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+2x+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+2x+\frac{5}{2}x+\frac{1}{2}\times 4$$

Combine 2x and \frac{5}{2}x to get \frac{9}{2}x.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+2$$

Subtract \frac{5}{2}x^{2} from both sides.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2-\frac{5}{2}x^{2}=\frac{9}{2}x+2$$

Combine \frac{5}{2}x^{2} and -\frac{5}{2}x^{2} to get 0.

$$\frac{9}{2}x+2=\frac{9}{2}x+2$$

Subtract \frac{9}{2}x from both sides.

$$\frac{9}{2}x+2-\frac{9}{2}x=2$$

Combine \frac{9}{2}x and -\frac{9}{2}x to get 0.

$$2=2$$

Compare 2 and 2.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}x by 5x-1.

$$\frac{1}{2}x\times 5x+\frac{1}{2}x\left(-1\right)+5x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

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

$$\frac{1}{2}x^{2}\times 5+\frac{1}{2}x\left(-1\right)+5x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Combine -\frac{1}{2}x and 5x to get \frac{9}{2}x.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5\left(x+1\right)-1\right)$$

Use the distributive property to multiply 5 by x+1.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5x+5-1\right)$$

Subtract 1 from 5 to get 4.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}\left(x+1\right)\left(5x+4\right)$$

Use the distributive property to multiply \frac{1}{2} by x+1.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\left(\frac{1}{2}x+\frac{1}{2}\right)\left(5x+4\right)$$

Apply the distributive property by multiplying each term of \frac{1}{2}x+\frac{1}{2} by each term of 5x+4.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}x\times 5x+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

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

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{1}{2}x^{2}\times 5+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{1}{2}x\times 4+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{4}{2}x+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Divide 4 by 2 to get 2.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+2x+\frac{1}{2}\times 5x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+2x+\frac{5}{2}x+\frac{1}{2}\times 4$$

Combine 2x and \frac{5}{2}x to get \frac{9}{2}x.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+\frac{1}{2}\times 4$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2=\frac{5}{2}x^{2}+\frac{9}{2}x+2$$

Subtract \frac{5}{2}x^{2} from both sides.

$$\frac{5}{2}x^{2}+\frac{9}{2}x+2-\frac{5}{2}x^{2}=\frac{9}{2}x+2$$

Combine \frac{5}{2}x^{2} and -\frac{5}{2}x^{2} to get 0.

$$\frac{9}{2}x+2=\frac{9}{2}x+2$$

Subtract \frac{9}{2}x from both sides.

$$\frac{9}{2}x+2-\frac{9}{2}x=2$$

Combine \frac{9}{2}x and -\frac{9}{2}x to get 0.

$$2=2$$

Compare 2 and 2.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$