Solve 3/4=1/2(2c+1/2c)

$$\frac { 3 } { 4 } = \frac { 1 } { 2 } ( 2 c + \frac { 1 } { 2 c } )$$

Solve for c

$$c=\frac{3+\sqrt{7}i}{8}\approx 0.375+0.330718914i$$

$$c=\frac{-\sqrt{7}i+3}{8}\approx 0.375-0.330718914i$$

Steps Using the Quadratic Formula

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$\frac{3}{4}\times 2=2c+\frac{1}{2c}$$

Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

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

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 2,2c.

$$3c=2c\times 2c+1$$

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

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

Multiply 2 and 2 to get 4.

$$3c=4c^{2}+1$$

Subtract 4c^{2} from both sides.

$$3c-4c^{2}=1$$

Subtract 1 from both sides.

$$3c-4c^{2}-1=0$$

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.

$$-4c^{2}+3c-1=0$$

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

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

Square 3.

$$c=\frac{-3±\sqrt{9-4\left(-4\right)\left(-1\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$c=\frac{-3±\sqrt{9+16\left(-1\right)}}{2\left(-4\right)}$$

Multiply 16 times -1.

$$c=\frac{-3±\sqrt{9-16}}{2\left(-4\right)}$$

Add 9 to -16.

$$c=\frac{-3±\sqrt{-7}}{2\left(-4\right)}$$

Take the square root of -7.

$$c=\frac{-3±\sqrt{7}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$c=\frac{-3±\sqrt{7}i}{-8}$$

Now solve the equation c=\frac{-3±\sqrt{7}i}{-8} when ± is plus. Add -3 to i\sqrt{7}.

$$c=\frac{-3+\sqrt{7}i}{-8}$$

Divide -3+i\sqrt{7} by -8.

$$c=\frac{-\sqrt{7}i+3}{8}$$

Now solve the equation c=\frac{-3±\sqrt{7}i}{-8} when ± is minus. Subtract i\sqrt{7} from -3.

$$c=\frac{-\sqrt{7}i-3}{-8}$$

Divide -3-i\sqrt{7} by -8.

$$c=\frac{3+\sqrt{7}i}{8}$$

The equation is now solved.

$$c=\frac{-\sqrt{7}i+3}{8} c=\frac{3+\sqrt{7}i}{8}$$

Steps for Completing the Square

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$\frac{3}{4}\times 2=2c+\frac{1}{2c}$$

Multiply \frac{3}{4} and 2 to get \frac{3}{2}.

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

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 2,2c.

$$3c=2c\times 2c+1$$

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

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

Multiply 2 and 2 to get 4.

$$3c=4c^{2}+1$$

Subtract 4c^{2} from both sides.

$$3c-4c^{2}=1$$

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.

$$-4c^{2}+3c=1$$

Divide both sides by -4.

$$\frac{-4c^{2}+3c}{-4}=\frac{1}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$c^{2}+\frac{3}{-4}c=\frac{1}{-4}$$

Divide 3 by -4.

$$c^{2}-\frac{3}{4}c=\frac{1}{-4}$$

Divide 1 by -4.

$$c^{2}-\frac{3}{4}c=-\frac{1}{4}$$

Divide -\frac{3}{4}, the coefficient of the x term, by 2 to get -\frac{3}{8}. Then add the square of -\frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}-\frac{3}{4}c+\left(-\frac{3}{8}\right)^{2}=-\frac{1}{4}+\left(-\frac{3}{8}\right)^{2}$$

Square -\frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$c^{2}-\frac{3}{4}c+\frac{9}{64}=-\frac{1}{4}+\frac{9}{64}$$

Add -\frac{1}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$c^{2}-\frac{3}{4}c+\frac{9}{64}=-\frac{7}{64}$$

Factor c^{2}-\frac{3}{4}c+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c-\frac{3}{8}\right)^{2}=-\frac{7}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-\frac{3}{8}\right)^{2}}=\sqrt{-\frac{7}{64}}$$

Simplify.

$$c-\frac{3}{8}=\frac{\sqrt{7}i}{8} c-\frac{3}{8}=-\frac{\sqrt{7}i}{8}$$

Add \frac{3}{8} to both sides of the equation.

$$c=\frac{3+\sqrt{7}i}{8} c=\frac{-\sqrt{7}i+3}{8}$$