Find all values of $c$ such that $\dfrac{c}{c-5} = \dfrac{4}{c-4}$. If you find more than one solution, then list the solutions you find separated by commas.

Guest Jan 21, 2018

#1**+1 **

Solve for c:

c/(c - 5) = 4/(c - 4)

Multiply both sides by a polynomial to clear fractions.

Cross multiply:

c (c - 4) = 4 (c - 5)

Write the quadratic polynomial on the left hand side in standard form.

Expand out terms of the left hand side:

c^2 - 4 c = 4 (c - 5)

Write the linear polynomial on the right hand side in standard form.

Expand out terms of the right hand side:

c^2 - 4 c = 4 c - 20

Move everything to the left hand side.

Subtract 4 c - 20 from both sides:

c^2 - 8 c + 20 = 0

Solve the quadratic equation by completing the square.

Subtract 20 from both sides:

c^2 - 8 c = -20

Take one half of the coefficient of c and square it, then add it to both sides.

Add 16 to both sides:

c^2 - 8 c + 16 = -4

Factor the left hand side.

Write the left hand side as a square:

(c - 4)^2 = -4

Eliminate the exponent on the left hand side.

Take the square root of both sides:

c - 4 = 2 i or c - 4 = -2 i

Look at the first equation: Solve for c.

Add 4 to both sides:

c = 4 + 2 i or c - 4 = -2 i

Look at the second equation: Solve for c.

Add 4 to both sides:

**c = 4 + 2 i or c = 4 - 2 i**

Guest Jan 21, 2018

#2**+1 **

c/(c-5) = 4/(c-4) cross multiply to get

c^2 - 4c = 4c-20 simplify to get

c^2 -8c + 20 = 0 Use Quadratic Equation

c = 4 +- sqrt (-16)/2

c= 4+- 2i

ElectricPavlov Jan 21, 2018