Solve c/5c-30-6/c^2-12c+36=c-5/6c-36

$$\frac { c } { 5 c - 30 } - \frac { 6 } { c ^ { 2 } - 12 c + 36 } = \frac { c - 5 } { 6 c - 36 }$$

Solve for c

$$c=-30$$

$$c=11$$

Steps Using Factoring

Variable c cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 30\left(c-6\right)^{2}, the least common multiple of 5c-30,c^{2}-12c+36,6c-36.

$$\left(6c-36\right)c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 6c-36 by c.

$$6c^{2}-36c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Multiply -30 and 6 to get -180.

$$6c^{2}-36c-180=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 5c-30 by c-5 and combine like terms.

$$6c^{2}-36c-180=5c^{2}-55c+150$$

Subtract 5c^{2} from both sides.

$$6c^{2}-36c-180-5c^{2}=-55c+150$$

Combine 6c^{2} and -5c^{2} to get c^{2}.

$$c^{2}-36c-180=-55c+150$$

Add 55c to both sides.

$$c^{2}-36c-180+55c=150$$

Combine -36c and 55c to get 19c.

$$c^{2}+19c-180=150$$

Subtract 150 from both sides.

$$c^{2}+19c-180-150=0$$

Subtract 150 from -180 to get -330.

$$c^{2}+19c-330=0$$

To solve the equation, factor c^{2}+19c-330 using formula c^{2}+\left(a+b\right)c+ab=\left(c+a\right)\left(c+b\right). To find a and b, set up a system to be solved.

$$a+b=19 ab=-330$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -330.

$$-1,330 -2,165 -3,110 -5,66 -6,55 -10,33 -11,30 -15,22$$

Calculate the sum for each pair.

$$-1+330=329 -2+165=163 -3+110=107 -5+66=61 -6+55=49 -10+33=23 -11+30=19 -15+22=7$$

The solution is the pair that gives sum 19.

$$a=-11 b=30$$

Rewrite factored expression \left(c+a\right)\left(c+b\right) using the obtained values.

$$\left(c-11\right)\left(c+30\right)$$

To find equation solutions, solve c-11=0 and c+30=0.

$$c=11 c=-30$$

Steps Using Factoring By Grouping

Variable c cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 30\left(c-6\right)^{2}, the least common multiple of 5c-30,c^{2}-12c+36,6c-36.

$$\left(6c-36\right)c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 6c-36 by c.

$$6c^{2}-36c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Multiply -30 and 6 to get -180.

$$6c^{2}-36c-180=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 5c-30 by c-5 and combine like terms.

$$6c^{2}-36c-180=5c^{2}-55c+150$$

Subtract 5c^{2} from both sides.

$$6c^{2}-36c-180-5c^{2}=-55c+150$$

Combine 6c^{2} and -5c^{2} to get c^{2}.

$$c^{2}-36c-180=-55c+150$$

Add 55c to both sides.

$$c^{2}-36c-180+55c=150$$

Combine -36c and 55c to get 19c.

$$c^{2}+19c-180=150$$

Subtract 150 from both sides.

$$c^{2}+19c-180-150=0$$

Subtract 150 from -180 to get -330.

$$c^{2}+19c-330=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as c^{2}+ac+bc-330. To find a and b, set up a system to be solved.

$$a+b=19 ab=1\left(-330\right)=-330$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -330.

$$-1,330 -2,165 -3,110 -5,66 -6,55 -10,33 -11,30 -15,22$$

Calculate the sum for each pair.

$$-1+330=329 -2+165=163 -3+110=107 -5+66=61 -6+55=49 -10+33=23 -11+30=19 -15+22=7$$

The solution is the pair that gives sum 19.

$$a=-11 b=30$$

Rewrite c^{2}+19c-330 as \left(c^{2}-11c\right)+\left(30c-330\right).

$$\left(c^{2}-11c\right)+\left(30c-330\right)$$

Factor out c in the first and 30 in the second group.

$$c\left(c-11\right)+30\left(c-11\right)$$

Factor out common term c-11 by using distributive property.

$$\left(c-11\right)\left(c+30\right)$$

To find equation solutions, solve c-11=0 and c+30=0.

$$c=11 c=-30$$

Steps Using the Quadratic Formula

Variable c cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 30\left(c-6\right)^{2}, the least common multiple of 5c-30,c^{2}-12c+36,6c-36.

$$\left(6c-36\right)c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 6c-36 by c.

$$6c^{2}-36c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Multiply -30 and 6 to get -180.

$$6c^{2}-36c-180=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 5c-30 by c-5 and combine like terms.

$$6c^{2}-36c-180=5c^{2}-55c+150$$

Subtract 5c^{2} from both sides.

$$6c^{2}-36c-180-5c^{2}=-55c+150$$

Combine 6c^{2} and -5c^{2} to get c^{2}.

$$c^{2}-36c-180=-55c+150$$

Add 55c to both sides.

$$c^{2}-36c-180+55c=150$$

Combine -36c and 55c to get 19c.

$$c^{2}+19c-180=150$$

Subtract 150 from both sides.

$$c^{2}+19c-180-150=0$$

Subtract 150 from -180 to get -330.

$$c^{2}+19c-330=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and -330 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-19±\sqrt{19^{2}-4\left(-330\right)}}{2}$$

Square 19.

$$c=\frac{-19±\sqrt{361-4\left(-330\right)}}{2}$$

Multiply -4 times -330.

$$c=\frac{-19±\sqrt{361+1320}}{2}$$

Add 361 to 1320.

$$c=\frac{-19±\sqrt{1681}}{2}$$

Take the square root of 1681.

$$c=\frac{-19±41}{2}$$

Now solve the equation c=\frac{-19±41}{2} when ± is plus. Add -19 to 41.

$$c=\frac{22}{2}$$

Divide 22 by 2.

$$c=11$$

Now solve the equation c=\frac{-19±41}{2} when ± is minus. Subtract 41 from -19.

$$c=-\frac{60}{2}$$

Divide -60 by 2.

$$c=-30$$

The equation is now solved.

$$c=11 c=-30$$

Steps for Completing the Square

Variable c cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 30\left(c-6\right)^{2}, the least common multiple of 5c-30,c^{2}-12c+36,6c-36.

$$\left(6c-36\right)c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 6c-36 by c.

$$6c^{2}-36c-30\times 6=\left(5c-30\right)\left(c-5\right)$$

Multiply -30 and 6 to get -180.

$$6c^{2}-36c-180=\left(5c-30\right)\left(c-5\right)$$

Use the distributive property to multiply 5c-30 by c-5 and combine like terms.

$$6c^{2}-36c-180=5c^{2}-55c+150$$

Subtract 5c^{2} from both sides.

$$6c^{2}-36c-180-5c^{2}=-55c+150$$

Combine 6c^{2} and -5c^{2} to get c^{2}.

$$c^{2}-36c-180=-55c+150$$

Add 55c to both sides.

$$c^{2}-36c-180+55c=150$$

Combine -36c and 55c to get 19c.

$$c^{2}+19c-180=150$$

Add 180 to both sides.

$$c^{2}+19c=150+180$$

Add 150 and 180 to get 330.

$$c^{2}+19c=330$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}+19c+\left(\frac{19}{2}\right)^{2}=330+\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$c^{2}+19c+\frac{361}{4}=330+\frac{361}{4}$$

Add 330 to \frac{361}{4}.

$$c^{2}+19c+\frac{361}{4}=\frac{1681}{4}$$

Factor c^{2}+19c+\frac{361}{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}.

$$\left(c+\frac{19}{2}\right)^{2}=\frac{1681}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+\frac{19}{2}\right)^{2}}=\sqrt{\frac{1681}{4}}$$

Simplify.

$$c+\frac{19}{2}=\frac{41}{2} c+\frac{19}{2}=-\frac{41}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$c=11 c=-30$$