Solve c^2=4c/5-36

$$c ^ { 2 } = \frac { 4 c } { 5 } - 36$$

Solve for c

$$c=\frac{2+8\sqrt{14}i}{5}\approx 0.4+5.986651819i$$

$$c=\frac{-8\sqrt{14}i+2}{5}\approx 0.4-5.986651819i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$5c^{2}=4c-180$$

Subtract 4c from both sides.

$$5c^{2}-4c=-180$$

Add 180 to both sides.

$$5c^{2}-4c+180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -4 for b, and 180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 5\times 180}}{2\times 5}$$

Square -4.

$$c=\frac{-\left(-4\right)±\sqrt{16-4\times 5\times 180}}{2\times 5}$$

Multiply -4 times 5.

$$c=\frac{-\left(-4\right)±\sqrt{16-20\times 180}}{2\times 5}$$

Multiply -20 times 180.

$$c=\frac{-\left(-4\right)±\sqrt{16-3600}}{2\times 5}$$

Add 16 to -3600.

$$c=\frac{-\left(-4\right)±\sqrt{-3584}}{2\times 5}$$

Take the square root of -3584.

$$c=\frac{-\left(-4\right)±16\sqrt{14}i}{2\times 5}$$

The opposite of -4 is 4.

$$c=\frac{4±16\sqrt{14}i}{2\times 5}$$

Multiply 2 times 5.

$$c=\frac{4±16\sqrt{14}i}{10}$$

Now solve the equation c=\frac{4±16\sqrt{14}i}{10} when ± is plus. Add 4 to 16i\sqrt{14}.

$$c=\frac{4+16\sqrt{14}i}{10}$$

Divide 4+16i\sqrt{14} by 10.

$$c=\frac{2+8\sqrt{14}i}{5}$$

Now solve the equation c=\frac{4±16\sqrt{14}i}{10} when ± is minus. Subtract 16i\sqrt{14} from 4.

$$c=\frac{-16\sqrt{14}i+4}{10}$$

Divide 4-16i\sqrt{14} by 10.

$$c=\frac{-8\sqrt{14}i+2}{5}$$

The equation is now solved.

$$c=\frac{2+8\sqrt{14}i}{5} c=\frac{-8\sqrt{14}i+2}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$5c^{2}=4c-180$$

Subtract 4c from both sides.

$$5c^{2}-4c=-180$$

Divide both sides by 5.

$$\frac{5c^{2}-4c}{5}=-\frac{180}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$c^{2}-\frac{4}{5}c=-\frac{180}{5}$$

Divide -180 by 5.

$$c^{2}-\frac{4}{5}c=-36$$

Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}-\frac{4}{5}c+\left(-\frac{2}{5}\right)^{2}=-36+\left(-\frac{2}{5}\right)^{2}$$

Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$c^{2}-\frac{4}{5}c+\frac{4}{25}=-36+\frac{4}{25}$$

Add -36 to \frac{4}{25}.

$$c^{2}-\frac{4}{5}c+\frac{4}{25}=-\frac{896}{25}$$

Factor c^{2}-\frac{4}{5}c+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c-\frac{2}{5}\right)^{2}=-\frac{896}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-\frac{2}{5}\right)^{2}}=\sqrt{-\frac{896}{25}}$$

Simplify.

$$c-\frac{2}{5}=\frac{8\sqrt{14}i}{5} c-\frac{2}{5}=-\frac{8\sqrt{14}i}{5}$$

Add \frac{2}{5} to both sides of the equation.

$$c=\frac{2+8\sqrt{14}i}{5} c=\frac{-8\sqrt{14}i+2}{5}$$