Solve frac{2sqrt{7}}{7}=c^2+74/2sqrt{7c}

$$\frac { 2 \sqrt { 7 } } { 7 } = \frac { c ^ { 2 } + 74 } { 2 \sqrt { 7 } c }$$

Solve for c

$$c=2+\sqrt{70}i\approx 2+8.366600265i$$

$$c=-\sqrt{70}i+2\approx 2-8.366600265i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 7.

$$2\sqrt{7}=\frac{1}{2}c^{-1}\times 7^{\frac{1}{2}}\left(c^{2}+74\right)$$

Use the distributive property to multiply \frac{1}{2}c^{-1}\times 7^{\frac{1}{2}} by c^{2}+74.

$$2\sqrt{7}=\frac{1}{2}c^{-1}\times 7^{\frac{1}{2}}c^{2}+37\times 7^{\frac{1}{2}}c^{-1}$$

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

$$2\sqrt{7}=\frac{1}{2}c^{1}\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}$$

Calculate c to the power of 1 and get c.

$$2\sqrt{7}=\frac{1}{2}c\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}$$

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

$$\frac{1}{2}c\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}=2\sqrt{7}$$

Subtract 2\sqrt{7} from both sides.

$$\frac{1}{2}c\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}-2\sqrt{7}=0$$

Reorder the terms.

$$\frac{1}{2}\sqrt{7}c-2\sqrt{7}+37\sqrt{7}\times \frac{1}{c}=0$$

Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2c, the least common multiple of 2,c.

$$\frac{1}{2}\sqrt{7}c\times 2c-2\sqrt{7}\times 2c+37\sqrt{7}\times 2\times 1=0$$

Multiply \frac{1}{2} and 2 to get 1.

$$\sqrt{7}cc-2\sqrt{7}\times 2c+37\sqrt{7}\times 2\times 1=0$$

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

$$\sqrt{7}c^{2}-2\sqrt{7}\times 2c+37\sqrt{7}\times 2\times 1=0$$

Multiply -2 and 2 to get -4.

$$\sqrt{7}c^{2}-4\sqrt{7}c+37\sqrt{7}\times 2\times 1=0$$

Multiply 37 and 2 to get 74.

$$\sqrt{7}c^{2}-4\sqrt{7}c+74\sqrt{7}\times 1=0$$

Multiply 74 and 1 to get 74.

$$\sqrt{7}c^{2}-4\sqrt{7}c+74\sqrt{7}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\sqrt{7}c^{2}+\left(-4\sqrt{7}\right)c+74\sqrt{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \sqrt{7} for a, -4\sqrt{7} for b, and 74\sqrt{7} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-4\sqrt{7}\right)±\sqrt{\left(-4\sqrt{7}\right)^{2}-4\sqrt{7}\times 74\sqrt{7}}}{2\sqrt{7}}$$

Square -4\sqrt{7}.

$$c=\frac{-\left(-4\sqrt{7}\right)±\sqrt{112-4\sqrt{7}\times 74\sqrt{7}}}{2\sqrt{7}}$$

Multiply -4 times \sqrt{7}.

$$c=\frac{-\left(-4\sqrt{7}\right)±\sqrt{112+\left(-4\sqrt{7}\right)\times 74\sqrt{7}}}{2\sqrt{7}}$$

Multiply -4\sqrt{7} times 74\sqrt{7}.

$$c=\frac{-\left(-4\sqrt{7}\right)±\sqrt{112-2072}}{2\sqrt{7}}$$

Add 112 to -2072.

$$c=\frac{-\left(-4\sqrt{7}\right)±\sqrt{-1960}}{2\sqrt{7}}$$

Take the square root of -1960.

$$c=\frac{-\left(-4\sqrt{7}\right)±14\sqrt{10}i}{2\sqrt{7}}$$

The opposite of -4\sqrt{7} is 4\sqrt{7}.

$$c=\frac{4\sqrt{7}±14\sqrt{10}i}{2\sqrt{7}}$$

Now solve the equation c=\frac{4\sqrt{7}±14\sqrt{10}i}{2\sqrt{7}} when ± is plus. Add 4\sqrt{7} to 14i\sqrt{10}.

$$c=\frac{4\sqrt{7}+14\sqrt{10}i}{2\sqrt{7}}$$

Divide 4\sqrt{7}+14i\sqrt{10} by 2\sqrt{7}.

$$c=2+\sqrt{70}i$$

Now solve the equation c=\frac{4\sqrt{7}±14\sqrt{10}i}{2\sqrt{7}} when ± is minus. Subtract 14i\sqrt{10} from 4\sqrt{7}.

$$c=\frac{-14\sqrt{10}i+4\sqrt{7}}{2\sqrt{7}}$$

Divide 4\sqrt{7}-14i\sqrt{10} by 2\sqrt{7}.

$$c=-\sqrt{70}i+2$$

The equation is now solved.

$$c=2+\sqrt{70}i c=-\sqrt{70}i+2$$

Steps for Completing the Square

Multiply both sides of the equation by 7.

$$2\sqrt{7}=\frac{1}{2}c^{-1}\times 7^{\frac{1}{2}}\left(c^{2}+74\right)$$

Use the distributive property to multiply \frac{1}{2}c^{-1}\times 7^{\frac{1}{2}} by c^{2}+74.

$$2\sqrt{7}=\frac{1}{2}c^{-1}\times 7^{\frac{1}{2}}c^{2}+37\times 7^{\frac{1}{2}}c^{-1}$$

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

$$2\sqrt{7}=\frac{1}{2}c^{1}\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}$$

Calculate c to the power of 1 and get c.

$$2\sqrt{7}=\frac{1}{2}c\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}$$

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

$$\frac{1}{2}c\times 7^{\frac{1}{2}}+37\times 7^{\frac{1}{2}}c^{-1}=2\sqrt{7}$$

Reorder the terms.

$$\frac{1}{2}\sqrt{7}c+37\sqrt{7}\times \frac{1}{c}=2\sqrt{7}$$

Variable c cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2c, the least common multiple of 2,c.

$$\frac{1}{2}\sqrt{7}c\times 2c+37\sqrt{7}\times 2\times 1=2\sqrt{7}\times 2c$$

Multiply \frac{1}{2} and 2 to get 1.

$$\sqrt{7}cc+37\sqrt{7}\times 2\times 1=2\sqrt{7}\times 2c$$

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

$$\sqrt{7}c^{2}+37\sqrt{7}\times 2\times 1=2\sqrt{7}\times 2c$$

Multiply 37 and 2 to get 74.

$$\sqrt{7}c^{2}+74\sqrt{7}\times 1=2\sqrt{7}\times 2c$$

Multiply 74 and 1 to get 74.

$$\sqrt{7}c^{2}+74\sqrt{7}=2\sqrt{7}\times 2c$$

Multiply 2 and 2 to get 4.

$$\sqrt{7}c^{2}+74\sqrt{7}=4\sqrt{7}c$$

Subtract 4\sqrt{7}c from both sides.

$$\sqrt{7}c^{2}+74\sqrt{7}-4\sqrt{7}c=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\sqrt{7}c^{2}+\left(-4\sqrt{7}\right)c+74\sqrt{7}=0$$

Subtract 74\sqrt{7} from both sides of the equation.

$$\sqrt{7}c^{2}+\left(-4\sqrt{7}\right)c+74\sqrt{7}-74\sqrt{7}=-74\sqrt{7}$$

Subtracting 74\sqrt{7} from itself leaves 0.

$$\sqrt{7}c^{2}+\left(-4\sqrt{7}\right)c=-74\sqrt{7}$$

Divide both sides by \sqrt{7}.

$$\frac{\sqrt{7}c^{2}+\left(-4\sqrt{7}\right)c}{\sqrt{7}}=-\frac{74\sqrt{7}}{\sqrt{7}}$$

Dividing by \sqrt{7} undoes the multiplication by \sqrt{7}.

$$c^{2}+\left(-\frac{4\sqrt{7}}{\sqrt{7}}\right)c=-\frac{74\sqrt{7}}{\sqrt{7}}$$

Divide -4\sqrt{7} by \sqrt{7}.

$$c^{2}-4c=-\frac{74\sqrt{7}}{\sqrt{7}}$$

Divide -74\sqrt{7} by \sqrt{7}.

$$c^{2}-4c=-74$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}-4c+\left(-2\right)^{2}=-74+\left(-2\right)^{2}$$

Square -2.

$$c^{2}-4c+4=-74+4$$

Add -74 to 4.

$$c^{2}-4c+4=-70$$

Factor c^{2}-4c+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}.

$$\left(c-2\right)^{2}=-70$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-2\right)^{2}}=\sqrt{-70}$$

Simplify.

$$c-2=\sqrt{70}i c-2=-\sqrt{70}i$$

Add 2 to both sides of the equation.

$$c=2+\sqrt{70}i c=-\sqrt{70}i+2$$