Solve x=sqrt{4^2+(8x)^2}

$$x = \sqrt { 4 ^ { 2 } + ( 8 x ) ^ { 2 } }$$

Solve for x (complex solution)

$$x=\frac{4\sqrt{7}i}{21}\approx 0.503952631i$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{4^{2}+\left(8x\right)^{2}}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x^{2}=\left(\sqrt{16+\left(8x\right)^{2}}\right)^{2}$$

Expand \left(8x\right)^{2}.

$$x^{2}=\left(\sqrt{16+8^{2}x^{2}}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$x^{2}=\left(\sqrt{16+64x^{2}}\right)^{2}$$

Calculate \sqrt{16+64x^{2}} to the power of 2 and get 16+64x^{2}.

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

Subtract 64x^{2} from both sides.

$$x^{2}-64x^{2}=16$$

Combine x^{2} and -64x^{2} to get -63x^{2}.

$$-63x^{2}=16$$

Divide both sides by -63.

$$x^{2}=-\frac{16}{63}$$

The equation is now solved.

$$x=\frac{4\sqrt{7}i}{21} x=-\frac{4\sqrt{7}i}{21}$$

Substitute \frac{4\sqrt{7}i}{21} for x in the equation x=\sqrt{4^{2}+\left(8x\right)^{2}}.

$$\frac{4\sqrt{7}i}{21}=\sqrt{4^{2}+\left(8\times \frac{4\sqrt{7}i}{21}\right)^{2}}$$

Simplify. The value x=\frac{4\sqrt{7}i}{21} satisfies the equation.

$$\frac{4}{21}i\times 7^{\frac{1}{2}}=\frac{4}{21}i\times 7^{\frac{1}{2}}$$

Substitute -\frac{4\sqrt{7}i}{21} for x in the equation x=\sqrt{4^{2}+\left(8x\right)^{2}}.

$$-\frac{4\sqrt{7}i}{21}=\sqrt{4^{2}+\left(8\left(-\frac{4\sqrt{7}i}{21}\right)\right)^{2}}$$

Simplify. The value x=-\frac{4\sqrt{7}i}{21} does not satisfy the equation.

$$-\frac{4}{21}i\times 7^{\frac{1}{2}}=\frac{4}{21}i\times 7^{\frac{1}{2}}$$

Equation x=\sqrt{\left(8x\right)^{2}+16} has a unique solution.

$$x=\frac{4\sqrt{7}i}{21}$$