Solve x^2+xsqrt{x^2-1}=2

$$x ^ { 2 } + x \sqrt { x ^ { 2 } - 1 } = 2$$

Solve for x

$$x = \frac{2 \sqrt{3}}{3} \approx 1.154700538$$

Solution Steps

Subtract x^{2} from both sides of the equation.

$$x\sqrt{x^{2}-1}=2-x^{2}$$

Square both sides of the equation.

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

Expand \left(x\sqrt{x^{2}-1}\right)^{2}.

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

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

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

Use the distributive property to multiply x^{2} by x^{2}-1.

$$x^{4}-x^{2}=\left(2-x^{2}\right)^{2}$$

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

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-x^{2}=4-4x^{2}+x^{4}$$

Add 4x^{2} to both sides.

$$x^{4}-x^{2}+4x^{2}=4+x^{4}$$

Combine -x^{2} and 4x^{2} to get 3x^{2}.

$$x^{4}+3x^{2}=4+x^{4}$$

Subtract x^{4} from both sides.

$$x^{4}+3x^{2}-x^{4}=4$$

Combine x^{4} and -x^{4} to get 0.

$$3x^{2}=4$$

Divide both sides by 3.

$$x^{2}=\frac{4}{3}$$

Take the square root of both sides of the equation.

$$x=\frac{2\sqrt{3}}{3} x=-\frac{2\sqrt{3}}{3}$$

Substitute \frac{2\sqrt{3}}{3} for x in the equation x^{2}+x\sqrt{x^{2}-1}=2.

$$\left(\frac{2\sqrt{3}}{3}\right)^{2}+\frac{2\sqrt{3}}{3}\sqrt{\left(\frac{2\sqrt{3}}{3}\right)^{2}-1}=2$$

Simplify. The value x=\frac{2\sqrt{3}}{3} satisfies the equation.

$$2=2$$

Substitute -\frac{2\sqrt{3}}{3} for x in the equation x^{2}+x\sqrt{x^{2}-1}=2.

$$\left(-\frac{2\sqrt{3}}{3}\right)^{2}+\left(-\frac{2\sqrt{3}}{3}\right)\sqrt{\left(-\frac{2\sqrt{3}}{3}\right)^{2}-1}=2$$

Simplify. The value x=-\frac{2\sqrt{3}}{3} does not satisfy the equation.

$$\frac{2}{3}=2$$

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

$$x=\frac{2\sqrt{3}}{3}$$