Solve (8+x)^2+x^2=16^22t

$$( 8 + x ) ^ { 2 } + x ^ { 2 } = 16 ^ { 2 } 2 t$$

Solve for t

$$t=\frac{x^{2}+8x+32}{256}$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8+x\right)^{2}.

$$64+16x+x^{2}+x^{2}=16^{2}\times 2t$$

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

$$64+16x+2x^{2}=16^{2}\times 2t$$

Calculate 16 to the power of 2 and get 256.

$$64+16x+2x^{2}=256\times 2t$$

Multiply 256 and 2 to get 512.

$$64+16x+2x^{2}=512t$$

Swap sides so that all variable terms are on the left hand side.

$$512t=64+16x+2x^{2}$$

The equation is in standard form.

$$512t=2x^{2}+16x+64$$

Divide both sides by 512.

$$\frac{512t}{512}=\frac{2x^{2}+16x+64}{512}$$

Dividing by 512 undoes the multiplication by 512.

$$t=\frac{2x^{2}+16x+64}{512}$$

Divide 64+16x+2x^{2} by 512.

$$t=\frac{x^{2}}{256}+\frac{x}{32}+\frac{1}{8}$$

Solve for x (complex solution)

$$x=4\sqrt{16t-1}-4$$

$$x=-4\sqrt{16t-1}-4$$

Solve for x

$$x=4\sqrt{16t-1}-4$$

$$x=-4\sqrt{16t-1}-4\text{, }t\geq \frac{1}{16}$$