Solve (1/2x)^2+6=x^2

$$( \frac { 1 } { 2 } x ) ^ { 2 } + 6 = x ^ { 2 }$$

Solve for x

$$x=2\sqrt{2}\approx 2.828427125$$

$$x=-2\sqrt{2}\approx -2.828427125$$

Steps by Finding Square Root

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

$$\left(\frac{1}{2}\right)^{2}x^{2}+6=x^{2}$$

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

$$\frac{1}{4}x^{2}+6=x^{2}$$

Subtract x^{2} from both sides.

$$\frac{1}{4}x^{2}+6-x^{2}=0$$

Combine \frac{1}{4}x^{2} and -x^{2} to get -\frac{3}{4}x^{2}.

$$-\frac{3}{4}x^{2}+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{3}{4}x^{2}=-6$$

Multiply both sides by -\frac{4}{3}, the reciprocal of -\frac{3}{4}.

$$x^{2}=-6\left(-\frac{4}{3}\right)$$

Multiply -6 and -\frac{4}{3} to get 8.

$$x^{2}=8$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

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

$$\left(\frac{1}{2}\right)^{2}x^{2}+6=x^{2}$$

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

$$\frac{1}{4}x^{2}+6=x^{2}$$

Subtract x^{2} from both sides.

$$\frac{1}{4}x^{2}+6-x^{2}=0$$

Combine \frac{1}{4}x^{2} and -x^{2} to get -\frac{3}{4}x^{2}.

$$-\frac{3}{4}x^{2}+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{3}{4} for a, 0 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{3}{4}\right)\times 6}}{2\left(-\frac{3}{4}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{3}{4}\right)\times 6}}{2\left(-\frac{3}{4}\right)}$$

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

$$x=\frac{0±\sqrt{3\times 6}}{2\left(-\frac{3}{4}\right)}$$

Multiply 3 times 6.

$$x=\frac{0±\sqrt{18}}{2\left(-\frac{3}{4}\right)}$$

Take the square root of 18.

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

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

$$x=\frac{0±3\sqrt{2}}{-\frac{3}{2}}$$

Now solve the equation x=\frac{0±3\sqrt{2}}{-\frac{3}{2}} when ± is plus.

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

Now solve the equation x=\frac{0±3\sqrt{2}}{-\frac{3}{2}} when ± is minus.

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

The equation is now solved.

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