Solve {l}{(3x+7)^2-84x=(3x-7)^2}{text{L.H.S.}=(3x+7)^2}

$$\left. \begin{array}{l}{ ( 3 x + 7 ) ^ { 2 } - 84 x = ( 3 x - 7 ) ^ { 2 } }\\{ \text { L.H.S. } = ( 3 x + 7 ) ^ { 2 } }\end{array} \right.$$

Solve for x (complex solution)

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+7\right)^{2}.

$$9x^{2}+42x+49-84x=\left(3x-7\right)^{2}$$

Combine 42x and -84x to get -42x.

$$9x^{2}-42x+49=\left(3x-7\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-7\right)^{2}.

$$9x^{2}-42x+49=9x^{2}-42x+49$$

Subtract 9x^{2} from both sides.

$$9x^{2}-42x+49-9x^{2}=-42x+49$$

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

$$-42x+49=-42x+49$$

Add 42x to both sides.

$$-42x+49+42x=49$$

Combine -42x and 42x to get 0.

$$49=49$$

Compare 49 and 49.

$$\text{true}$$

This is true for any x.

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

Solve for x

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+7\right)^{2}.

$$9x^{2}+42x+49-84x=\left(3x-7\right)^{2}$$

Combine 42x and -84x to get -42x.

$$9x^{2}-42x+49=\left(3x-7\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-7\right)^{2}.

$$9x^{2}-42x+49=9x^{2}-42x+49$$

Subtract 9x^{2} from both sides.

$$9x^{2}-42x+49-9x^{2}=-42x+49$$

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

$$-42x+49=-42x+49$$

Add 42x to both sides.

$$-42x+49+42x=49$$

Combine -42x and 42x to get 0.

$$49=49$$

Compare 49 and 49.

$$\text{true}$$

This is true for any x.

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