Solve xsqrt{457}*35=sqrt{x^2+2}

$$x \sqrt{ 457 } \times 35= \sqrt{ { x }^{ 2 } +2 }$$

Solve for x

$$x=\frac{\sqrt{69978}}{139956}\approx 0.001890119$$

Solution Steps

Square both sides of the equation.

$$\left(35x\sqrt{457}\right)^{2}=\left(\sqrt{x^{2}+2}\right)^{2}$$

Expand \left(35x\sqrt{457}\right)^{2}.

$$35^{2}x^{2}\left(\sqrt{457}\right)^{2}=\left(\sqrt{x^{2}+2}\right)^{2}$$

Calculate 35 to the power of 2 and get 1225.

$$1225x^{2}\left(\sqrt{457}\right)^{2}=\left(\sqrt{x^{2}+2}\right)^{2}$$

The square of \sqrt{457} is 457.

$$1225x^{2}\times 457=\left(\sqrt{x^{2}+2}\right)^{2}$$

Multiply 1225 and 457 to get 559825.

$$559825x^{2}=\left(\sqrt{x^{2}+2}\right)^{2}$$

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

$$559825x^{2}=x^{2}+2$$

Subtract x^{2} from both sides.

$$559825x^{2}-x^{2}=2$$

Combine 559825x^{2} and -x^{2} to get 559824x^{2}.

$$559824x^{2}=2$$

Divide both sides by 559824.

$$x^{2}=\frac{2}{559824}$$

Reduce the fraction \frac{2}{559824} to lowest terms by extracting and canceling out 2.

$$x^{2}=\frac{1}{279912}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{69978}}{139956} x=-\frac{\sqrt{69978}}{139956}$$

Substitute \frac{\sqrt{69978}}{139956} for x in the equation 35x\sqrt{457}=\sqrt{x^{2}+2}.

$$35\times \frac{\sqrt{69978}}{139956}\sqrt{457}=\sqrt{\left(\frac{\sqrt{69978}}{139956}\right)^{2}+2}$$

Simplify. The value x=\frac{\sqrt{69978}}{139956} satisfies the equation.

$$\frac{35}{139956}\times 31979946^{\frac{1}{2}}=\frac{35}{139956}\times 31979946^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{69978}}{139956} for x in the equation 35x\sqrt{457}=\sqrt{x^{2}+2}.

$$35\left(-\frac{\sqrt{69978}}{139956}\right)\sqrt{457}=\sqrt{\left(-\frac{\sqrt{69978}}{139956}\right)^{2}+2}$$

Simplify. The value x=-\frac{\sqrt{69978}}{139956} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{35}{139956}\times 31979946^{\frac{1}{2}}=\frac{35}{139956}\times 31979946^{\frac{1}{2}}$$

Equation 35\sqrt{457}x=\sqrt{x^{2}+2} has a unique solution.

$$x=\frac{\sqrt{69978}}{139956}$$