Solve 1/2(x-1)(x-8)[(x-7)-(x-2)]

$$\frac { 1 } { 2 } ( x - 1 ) ( x - 8 ) [ ( x - 7 ) - ( x - 2 ) ]$$

Evaluate

$$\frac{5\left(1-x\right)\left(x-8\right)}{2}$$

Solution Steps

To find the opposite of x-2, find the opposite of each term.

$$\frac{1}{2}\left(x-1\right)\left(x-8\right)\left(x-7-x-\left(-2\right)\right)$$

The opposite of -2 is 2.

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

Combine x and -x to get 0.

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

Add -7 and 2 to get -5.

$$\frac{1}{2}\left(x-1\right)\left(x-8\right)\left(-5\right)$$

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

$$\frac{-5}{2}\left(x-1\right)\left(x-8\right)$$

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$-\frac{5}{2}\left(x-1\right)\left(x-8\right)$$

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

$$\left(-\frac{5}{2}x-\frac{5}{2}\left(-1\right)\right)\left(x-8\right)$$

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

$$\left(-\frac{5}{2}x+\frac{5}{2}\right)\left(x-8\right)$$

Apply the distributive property by multiplying each term of -\frac{5}{2}x+\frac{5}{2} by each term of x-8.

$$-\frac{5}{2}xx-\frac{5}{2}x\left(-8\right)+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

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

$$-\frac{5}{2}x^{2}-\frac{5}{2}x\left(-8\right)+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Express -\frac{5}{2}\left(-8\right) as a single fraction.

$$-\frac{5}{2}x^{2}+\frac{-5\left(-8\right)}{2}x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Multiply -5 and -8 to get 40.

$$-\frac{5}{2}x^{2}+\frac{40}{2}x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Divide 40 by 2 to get 20.

$$-\frac{5}{2}x^{2}+20x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Combine 20x and \frac{5}{2}x to get \frac{45}{2}x.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{5}{2}\left(-8\right)$$

Express \frac{5}{2}\left(-8\right) as a single fraction.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{5\left(-8\right)}{2}$$

Multiply 5 and -8 to get -40.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{-40}{2}$$

Divide -40 by 2 to get -20.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x-20$$

Expand

$$-\frac{5x^{2}}{2}+\frac{45x}{2}-20$$

Solution Steps

To find the opposite of x-2, find the opposite of each term.

$$\frac{1}{2}\left(x-1\right)\left(x-8\right)\left(x-7-x-\left(-2\right)\right)$$

The opposite of -2 is 2.

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

Combine x and -x to get 0.

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

Add -7 and 2 to get -5.

$$\frac{1}{2}\left(x-1\right)\left(x-8\right)\left(-5\right)$$

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

$$\frac{-5}{2}\left(x-1\right)\left(x-8\right)$$

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$-\frac{5}{2}\left(x-1\right)\left(x-8\right)$$

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

$$\left(-\frac{5}{2}x-\frac{5}{2}\left(-1\right)\right)\left(x-8\right)$$

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

$$\left(-\frac{5}{2}x+\frac{5}{2}\right)\left(x-8\right)$$

Apply the distributive property by multiplying each term of -\frac{5}{2}x+\frac{5}{2} by each term of x-8.

$$-\frac{5}{2}xx-\frac{5}{2}x\left(-8\right)+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

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

$$-\frac{5}{2}x^{2}-\frac{5}{2}x\left(-8\right)+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Express -\frac{5}{2}\left(-8\right) as a single fraction.

$$-\frac{5}{2}x^{2}+\frac{-5\left(-8\right)}{2}x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Multiply -5 and -8 to get 40.

$$-\frac{5}{2}x^{2}+\frac{40}{2}x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Divide 40 by 2 to get 20.

$$-\frac{5}{2}x^{2}+20x+\frac{5}{2}x+\frac{5}{2}\left(-8\right)$$

Combine 20x and \frac{5}{2}x to get \frac{45}{2}x.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{5}{2}\left(-8\right)$$

Express \frac{5}{2}\left(-8\right) as a single fraction.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{5\left(-8\right)}{2}$$

Multiply 5 and -8 to get -40.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x+\frac{-40}{2}$$

Divide -40 by 2 to get -20.

$$-\frac{5}{2}x^{2}+\frac{45}{2}x-20$$