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

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

Evaluate

$$7x^{2}-11x+16$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$9x^{2}-6x+1-\left(x+1\right)\left(x-1\right)-\left(x+7\right)\left(x-2\right)$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$9x^{2}-6x+1-\left(x^{2}-1\right)-\left(x+7\right)\left(x-2\right)$$

To find the opposite of x^{2}-1, find the opposite of each term.

$$9x^{2}-6x+1-x^{2}+1-\left(x+7\right)\left(x-2\right)$$

Combine 9x^{2} and -x^{2} to get 8x^{2}.

$$8x^{2}-6x+1+1-\left(x+7\right)\left(x-2\right)$$

Add 1 and 1 to get 2.

$$8x^{2}-6x+2-\left(x+7\right)\left(x-2\right)$$

Use the distributive property to multiply x+7 by x-2 and combine like terms.

$$8x^{2}-6x+2-\left(x^{2}+5x-14\right)$$

To find the opposite of x^{2}+5x-14, find the opposite of each term.

$$8x^{2}-6x+2-x^{2}-5x+14$$

Combine 8x^{2} and -x^{2} to get 7x^{2}.

$$7x^{2}-6x+2-5x+14$$

Combine -6x and -5x to get -11x.

$$7x^{2}-11x+2+14$$

Add 2 and 14 to get 16.

$$7x^{2}-11x+16$$

Expand

$$7x^{2}-11x+16$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$9x^{2}-6x+1-\left(x+1\right)\left(x-1\right)-\left(x+7\right)\left(x-2\right)$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$9x^{2}-6x+1-\left(x^{2}-1\right)-\left(x+7\right)\left(x-2\right)$$

To find the opposite of x^{2}-1, find the opposite of each term.

$$9x^{2}-6x+1-x^{2}+1-\left(x+7\right)\left(x-2\right)$$

Combine 9x^{2} and -x^{2} to get 8x^{2}.

$$8x^{2}-6x+1+1-\left(x+7\right)\left(x-2\right)$$

Add 1 and 1 to get 2.

$$8x^{2}-6x+2-\left(x+7\right)\left(x-2\right)$$

Use the distributive property to multiply x+7 by x-2 and combine like terms.

$$8x^{2}-6x+2-\left(x^{2}+5x-14\right)$$

To find the opposite of x^{2}+5x-14, find the opposite of each term.

$$8x^{2}-6x+2-x^{2}-5x+14$$

Combine 8x^{2} and -x^{2} to get 7x^{2}.

$$7x^{2}-6x+2-5x+14$$

Combine -6x and -5x to get -11x.

$$7x^{2}-11x+2+14$$

Add 2 and 14 to get 16.

$$7x^{2}-11x+16$$