Solve 8+(1+3a)^2=8a^2+(a+3)^2

$$8 + ( 1 + 3 a ) ^ { 2 } = 8 a ^ { 2 } + ( a + 3 ) ^ { 2 }$$

Solve for a (complex solution)

$$a\in \mathrm{C}$$

Solution Steps

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

$$8+1+6a+9a^{2}=8a^{2}+\left(a+3\right)^{2}$$

Add 8 and 1 to get 9.

$$9+6a+9a^{2}=8a^{2}+\left(a+3\right)^{2}$$

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

$$9+6a+9a^{2}=8a^{2}+a^{2}+6a+9$$

Combine 8a^{2} and a^{2} to get 9a^{2}.

$$9+6a+9a^{2}=9a^{2}+6a+9$$

Subtract 9a^{2} from both sides.

$$9+6a+9a^{2}-9a^{2}=6a+9$$

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

$$9+6a=6a+9$$

Subtract 6a from both sides.

$$9+6a-6a=9$$

Combine 6a and -6a to get 0.

$$9=9$$

Compare 9 and 9.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{C}$$

Solve for a

$$a\in \mathrm{R}$$

Solution Steps

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

$$8+1+6a+9a^{2}=8a^{2}+\left(a+3\right)^{2}$$

Add 8 and 1 to get 9.

$$9+6a+9a^{2}=8a^{2}+\left(a+3\right)^{2}$$

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

$$9+6a+9a^{2}=8a^{2}+a^{2}+6a+9$$

Combine 8a^{2} and a^{2} to get 9a^{2}.

$$9+6a+9a^{2}=9a^{2}+6a+9$$

Subtract 9a^{2} from both sides.

$$9+6a+9a^{2}-9a^{2}=6a+9$$

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

$$9+6a=6a+9$$

Subtract 6a from both sides.

$$9+6a-6a=9$$

Combine 6a and -6a to get 0.

$$9=9$$

Compare 9 and 9.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$