Solve c=33+5c^2/4+19c

$$c = 33 + \frac { 5 c ^ { 2 } } { 4 } + 19 c$$

Solve for c

$$c=\frac{2\sqrt{159}-36}{5}\approx -2.156191915$$

$$c=\frac{-2\sqrt{159}-36}{5}\approx -12.243808085$$

Steps Using the Quadratic Formula

Subtract 33 from both sides.

$$c-33=\frac{5c^{2}}{4}+19c$$

Subtract \frac{5c^{2}}{4} from both sides.

$$c-33-\frac{5c^{2}}{4}=19c$$

Subtract 19c from both sides.

$$c-33-\frac{5c^{2}}{4}-19c=0$$

Combine c and -19c to get -18c.

$$-18c-33-\frac{5c^{2}}{4}=0$$

Multiply both sides of the equation by 4.

$$-72c-132-5c^{2}=0$$

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.

$$-5c^{2}-72c-132=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -72 for b, and -132 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\left(-5\right)\left(-132\right)}}{2\left(-5\right)}$$

Square -72.

$$c=\frac{-\left(-72\right)±\sqrt{5184-4\left(-5\right)\left(-132\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$c=\frac{-\left(-72\right)±\sqrt{5184+20\left(-132\right)}}{2\left(-5\right)}$$

Multiply 20 times -132.

$$c=\frac{-\left(-72\right)±\sqrt{5184-2640}}{2\left(-5\right)}$$

Add 5184 to -2640.

$$c=\frac{-\left(-72\right)±\sqrt{2544}}{2\left(-5\right)}$$

Take the square root of 2544.

$$c=\frac{-\left(-72\right)±4\sqrt{159}}{2\left(-5\right)}$$

The opposite of -72 is 72.

$$c=\frac{72±4\sqrt{159}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$c=\frac{72±4\sqrt{159}}{-10}$$

Now solve the equation c=\frac{72±4\sqrt{159}}{-10} when ± is plus. Add 72 to 4\sqrt{159}.

$$c=\frac{4\sqrt{159}+72}{-10}$$

Divide 72+4\sqrt{159} by -10.

$$c=\frac{-2\sqrt{159}-36}{5}$$

Now solve the equation c=\frac{72±4\sqrt{159}}{-10} when ± is minus. Subtract 4\sqrt{159} from 72.

$$c=\frac{72-4\sqrt{159}}{-10}$$

Divide 72-4\sqrt{159} by -10.

$$c=\frac{2\sqrt{159}-36}{5}$$

The equation is now solved.

$$c=\frac{-2\sqrt{159}-36}{5} c=\frac{2\sqrt{159}-36}{5}$$

Steps for Completing the Square

Subtract \frac{5c^{2}}{4} from both sides.

$$c-\frac{5c^{2}}{4}=33+19c$$

Subtract 19c from both sides.

$$c-\frac{5c^{2}}{4}-19c=33$$

Combine c and -19c to get -18c.

$$-18c-\frac{5c^{2}}{4}=33$$

Multiply both sides of the equation by 4.

$$-72c-5c^{2}=132$$

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.

$$-5c^{2}-72c=132$$

Divide both sides by -5.

$$\frac{-5c^{2}-72c}{-5}=\frac{132}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$c^{2}+\left(-\frac{72}{-5}\right)c=\frac{132}{-5}$$

Divide -72 by -5.

$$c^{2}+\frac{72}{5}c=\frac{132}{-5}$$

Divide 132 by -5.

$$c^{2}+\frac{72}{5}c=-\frac{132}{5}$$

Divide \frac{72}{5}, the coefficient of the x term, by 2 to get \frac{36}{5}. Then add the square of \frac{36}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}+\frac{72}{5}c+\left(\frac{36}{5}\right)^{2}=-\frac{132}{5}+\left(\frac{36}{5}\right)^{2}$$

Square \frac{36}{5} by squaring both the numerator and the denominator of the fraction.

$$c^{2}+\frac{72}{5}c+\frac{1296}{25}=-\frac{132}{5}+\frac{1296}{25}$$

Add -\frac{132}{5} to \frac{1296}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$c^{2}+\frac{72}{5}c+\frac{1296}{25}=\frac{636}{25}$$

Factor c^{2}+\frac{72}{5}c+\frac{1296}{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}.

$$\left(c+\frac{36}{5}\right)^{2}=\frac{636}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+\frac{36}{5}\right)^{2}}=\sqrt{\frac{636}{25}}$$

Simplify.

$$c+\frac{36}{5}=\frac{2\sqrt{159}}{5} c+\frac{36}{5}=-\frac{2\sqrt{159}}{5}$$

Subtract \frac{36}{5} from both sides of the equation.

$$c=\frac{2\sqrt{159}-36}{5} c=\frac{-2\sqrt{159}-36}{5}$$