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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.

 Jan 21, 2018
 #1
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+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

 Jan 21, 2018
 #2
avatar+17331 
+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

 Jan 21, 2018

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