Solve 8(x)=sqrt{x^2-1}

$$8 ( x ) = \sqrt { x ^ { 2 } - 1 }$$

Solve for x (complex solution)

$$x=\frac{\sqrt{7}i}{21}\approx 0.125988158i$$

Solution Steps

Square both sides of the equation.

$$\left(8x\right)^{2}=\left(\sqrt{x^{2}-1}\right)^{2}$$

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

$$8^{2}x^{2}=\left(\sqrt{x^{2}-1}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$64x^{2}=\left(\sqrt{x^{2}-1}\right)^{2}$$

Calculate \sqrt{x^{2}-1} to the power of 2 and get x^{2}-1.

$$64x^{2}=x^{2}-1$$

Subtract x^{2} from both sides.

$$64x^{2}-x^{2}=-1$$

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

$$63x^{2}=-1$$

Divide both sides by 63.

$$x^{2}=-\frac{1}{63}$$

The equation is now solved.

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

Substitute \frac{\sqrt{7}i}{21} for x in the equation 8x=\sqrt{x^{2}-1}.

$$8\times \frac{\sqrt{7}i}{21}=\sqrt{\left(\frac{\sqrt{7}i}{21}\right)^{2}-1}$$

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

$$\frac{8}{21}i\times 7^{\frac{1}{2}}=\frac{8}{21}i\times 7^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{7}i}{21} for x in the equation 8x=\sqrt{x^{2}-1}.

$$8\left(-\frac{\sqrt{7}i}{21}\right)=\sqrt{\left(-\frac{\sqrt{7}i}{21}\right)^{2}-1}$$

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

$$-\frac{8}{21}i\times 7^{\frac{1}{2}}=\frac{8}{21}i\times 7^{\frac{1}{2}}$$

Equation 8x=\sqrt{x^{2}-1} has a unique solution.

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