Solve 1+c^2/2sqrt{3c}=frac{sqrt{1}}{2}

$$\frac { 1 + c ^ { 2 } } { 2 \sqrt { 3 } c } = \frac { \sqrt { 1 } } { 2 }$$

Solve for c

$$c=\frac{\sqrt{3}+i}{2}\approx 0.866025404+0.5i$$

$$c=\frac{\sqrt{3}-i}{2}\approx 0.866025404-0.5i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply \frac{1}{3}\times 3^{\frac{1}{2}}c^{-1} by 1+c^{2}.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}c^{2}=\sqrt{1}$$

To multiply powers of the same base, add their exponents. Add -1 and 2 to get 1.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c^{1}=\sqrt{1}$$

Calculate c to the power of 1 and get c.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c=\sqrt{1}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c=1$$

Subtract 1 from both sides.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c-1=0$$

Reorder the terms.

$$\frac{1}{3}\sqrt{3}c-1+\frac{1}{3}\sqrt{3}\times \frac{1}{c}=0$$

Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3c, the least common multiple of 3,c.

$$\frac{1}{3}\sqrt{3}c\times 3c+3c\left(-1\right)+\frac{1}{3}\sqrt{3}\times 3\times 1=0$$

Multiply \frac{1}{3} and 3 to get 1.

$$\sqrt{3}cc+3c\left(-1\right)+\frac{1}{3}\sqrt{3}\times 3\times 1=0$$

Multiply c and c to get c^{2}.

$$\sqrt{3}c^{2}+3c\left(-1\right)+\frac{1}{3}\sqrt{3}\times 3\times 1=0$$

Multiply 3 and -1 to get -3.

$$\sqrt{3}c^{2}-3c+\frac{1}{3}\sqrt{3}\times 3\times 1=0$$

Multiply \frac{1}{3} and 3 to get 1.

$$\sqrt{3}c^{2}-3c+\sqrt{3}\times 1=0$$

Reorder the terms.

$$\sqrt{3}c^{2}-3c+\sqrt{3}=0$$

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

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

Square -3.

$$c=\frac{-\left(-3\right)±\sqrt{9-4\sqrt{3}\sqrt{3}}}{2\sqrt{3}}$$

Multiply -4 times \sqrt{3}.

$$c=\frac{-\left(-3\right)±\sqrt{9+\left(-4\sqrt{3}\right)\sqrt{3}}}{2\sqrt{3}}$$

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

$$c=\frac{-\left(-3\right)±\sqrt{9-12}}{2\sqrt{3}}$$

Add 9 to -12.

$$c=\frac{-\left(-3\right)±\sqrt{-3}}{2\sqrt{3}}$$

Take the square root of -3.

$$c=\frac{-\left(-3\right)±\sqrt{3}i}{2\sqrt{3}}$$

The opposite of -3 is 3.

$$c=\frac{3±\sqrt{3}i}{2\sqrt{3}}$$

Now solve the equation c=\frac{3±\sqrt{3}i}{2\sqrt{3}} when ± is plus. Add 3 to i\sqrt{3}.

$$c=\frac{3+\sqrt{3}i}{2\sqrt{3}}$$

Divide 3+i\sqrt{3} by 2\sqrt{3}.

$$c=\frac{\sqrt{3}}{2}+\frac{1}{2}i$$

Now solve the equation c=\frac{3±\sqrt{3}i}{2\sqrt{3}} when ± is minus. Subtract i\sqrt{3} from 3.

$$c=\frac{-\sqrt{3}i+3}{2\sqrt{3}}$$

Divide 3-i\sqrt{3} by 2\sqrt{3}.

$$c=\frac{\sqrt{3}}{2}-\frac{1}{2}i$$

The equation is now solved.

$$c=\frac{\sqrt{3}}{2}+\frac{1}{2}i c=\frac{\sqrt{3}}{2}-\frac{1}{2}i$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply \frac{1}{3}\times 3^{\frac{1}{2}}c^{-1} by 1+c^{2}.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}c^{2}=\sqrt{1}$$

To multiply powers of the same base, add their exponents. Add -1 and 2 to get 1.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c^{1}=\sqrt{1}$$

Calculate c to the power of 1 and get c.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c=\sqrt{1}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{3}\times 3^{\frac{1}{2}}c^{-1}+\frac{1}{3}\times 3^{\frac{1}{2}}c=1$$

Reorder the terms.

$$\frac{1}{3}\sqrt{3}c+\frac{1}{3}\sqrt{3}\times \frac{1}{c}=1$$

Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3c, the least common multiple of 3,c.

$$\frac{1}{3}\sqrt{3}c\times 3c+\frac{1}{3}\sqrt{3}\times 3\times 1=3c$$

Multiply \frac{1}{3} and 3 to get 1.

$$\sqrt{3}cc+\frac{1}{3}\sqrt{3}\times 3\times 1=3c$$

Multiply c and c to get c^{2}.

$$\sqrt{3}c^{2}+\frac{1}{3}\sqrt{3}\times 3\times 1=3c$$

Multiply \frac{1}{3} and 3 to get 1.

$$\sqrt{3}c^{2}+\sqrt{3}\times 1=3c$$

Subtract 3c from both sides.

$$\sqrt{3}c^{2}+\sqrt{3}\times 1-3c=0$$

Subtract \sqrt{3}\times 1 from both sides. Anything subtracted from zero gives its negation.

$$\sqrt{3}c^{2}-3c=-\sqrt{3}\times 1$$

Reorder the terms.

$$\sqrt{3}c^{2}-3c=-\sqrt{3}$$

Divide both sides by \sqrt{3}.

$$\frac{\sqrt{3}c^{2}-3c}{\sqrt{3}}=-\frac{\sqrt{3}}{\sqrt{3}}$$

Dividing by \sqrt{3} undoes the multiplication by \sqrt{3}.

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

Divide -3 by \sqrt{3}.

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

Divide -\sqrt{3} by \sqrt{3}.

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

Divide -\sqrt{3}, the coefficient of the x term, by 2 to get -\frac{\sqrt{3}}{2}. Then add the square of -\frac{\sqrt{3}}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{\sqrt{3}}{2}.

$$c^{2}+\left(-\sqrt{3}\right)c+\frac{3}{4}=-1+\frac{3}{4}$$

Add -1 to \frac{3}{4}.

$$c^{2}+\left(-\sqrt{3}\right)c+\frac{3}{4}=-\frac{1}{4}$$

Factor c^{2}+\left(-\sqrt{3}\right)c+\frac{3}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c-\frac{\sqrt{3}}{2}\right)^{2}=-\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-\frac{\sqrt{3}}{2}\right)^{2}}=\sqrt{-\frac{1}{4}}$$

Simplify.

$$c-\frac{\sqrt{3}}{2}=\frac{1}{2}i c-\frac{\sqrt{3}}{2}=-\frac{1}{2}i$$

Add \frac{\sqrt{3}}{2} to both sides of the equation.

$$c=\frac{\sqrt{3}}{2}+\frac{1}{2}i c=\frac{\sqrt{3}}{2}-\frac{1}{2}i$$