Solve -(9x+6)^2+(3x+7)(3x-7)

$$- ( 9 x + 6 ) ^ { 2 } + ( 3 x + 7 ) ( 3 x - 7 )$$

Evaluate

$$-72x^{2}-108x-85$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+6\right)^{2}.

$$-\left(81x^{2}+108x+36\right)+\left(3x+7\right)\left(3x-7\right)$$

To find the opposite of 81x^{2}+108x+36, find the opposite of each term.

$$-81x^{2}-108x-36+\left(3x+7\right)\left(3x-7\right)$$

Consider \left(3x+7\right)\left(3x-7\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 7.

$$-81x^{2}-108x-36+\left(3x\right)^{2}-49$$

Expand \left(3x\right)^{2}.

$$-81x^{2}-108x-36+3^{2}x^{2}-49$$

Calculate 3 to the power of 2 and get 9.

$$-81x^{2}-108x-36+9x^{2}-49$$

Combine -81x^{2} and 9x^{2} to get -72x^{2}.

$$-72x^{2}-108x-36-49$$

Subtract 49 from -36 to get -85.

$$-72x^{2}-108x-85$$

Expand

$$-72x^{2}-108x-85$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9x+6\right)^{2}.

$$-\left(81x^{2}+108x+36\right)+\left(3x+7\right)\left(3x-7\right)$$

To find the opposite of 81x^{2}+108x+36, find the opposite of each term.

$$-81x^{2}-108x-36+\left(3x+7\right)\left(3x-7\right)$$

Consider \left(3x+7\right)\left(3x-7\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 7.

$$-81x^{2}-108x-36+\left(3x\right)^{2}-49$$

Expand \left(3x\right)^{2}.

$$-81x^{2}-108x-36+3^{2}x^{2}-49$$

Calculate 3 to the power of 2 and get 9.

$$-81x^{2}-108x-36+9x^{2}-49$$

Combine -81x^{2} and 9x^{2} to get -72x^{2}.

$$-72x^{2}-108x-36-49$$

Subtract 49 from -36 to get -85.

$$-72x^{2}-108x-85$$