Solve sqrt{{left(sqrt{2}right)}^2+{left(1/2right)}^2}=x^2

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

Solve for x

$$x = \frac{\sqrt{6}}{2} \approx 1.224744871$$

$$x = -\frac{\sqrt{6}}{2} \approx -1.224744871$$

Steps by Finding Square Root

The square of \sqrt{2} is 2.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\sqrt{2+\frac{1}{4}}=x^{2}$$

Add 2 and \frac{1}{4} to get \frac{9}{4}.

$$\sqrt{\frac{9}{4}}=x^{2}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

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

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

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

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{6}}{2} x=-\frac{\sqrt{6}}{2}$$

Steps Using the Quadratic Formula

The square of \sqrt{2} is 2.

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

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\sqrt{2+\frac{1}{4}}=x^{2}$$

Add 2 and \frac{1}{4} to get \frac{9}{4}.

$$\sqrt{\frac{9}{4}}=x^{2}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

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

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

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

Subtract \frac{3}{2} from both sides.

$$x^{2}-\frac{3}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{3}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{3}{2}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{3}{2}\right)}}{2}$$

Multiply -4 times -\frac{3}{2}.

$$x=\frac{0±\sqrt{6}}{2}$$

Now solve the equation x=\frac{0±\sqrt{6}}{2} when ± is plus.

$$x=\frac{\sqrt{6}}{2}$$

Now solve the equation x=\frac{0±\sqrt{6}}{2} when ± is minus.

$$x=-\frac{\sqrt{6}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{6}}{2} x=-\frac{\sqrt{6}}{2}$$