solve the equation: (m+2/m^2-8m+15)-(2/m^2-10m+25)=-(m-2/m^2-8m+15)

Question

can someone solves this one? I don't get it

Answer 1

First, factor the denominators:

m^2 - 8m + 15 = (m -3)(m -5)

m^2 -10m + 25 = (m -5)(m-5)

 

Next, we'll want to clear the denominators by multiplying by a factor that will cause all of the denominators to cancel out. For example, the we can clear the denominator of the first term (m + 2)/(m -3)(m -5) by multiplying by (m -3)(m -5)(m -5)/(m -3)(m -5)(m -5), which is equal to 1. For the entire equation, we'll multiply by as follows:

 

(m + 2)/(m -3)(m -5)(m -5)*(m -3)(m -5)(m -5)/(m -3)(m -5) - (2/(m -5)(m-5))*(m -3)(m -5)(m-5)/(m -5)(m-5)(m -3) = -(m - 2)/(m -3)(m -5)(m -5)*(m -3)(m -5)/(m -3)(m -5)(m -5)

 

This simplifies to:

 

(m + 2)(m -5) - 2(m - 3) = -(m - 2)(m -5)

 

Foil and distribute constants: m^2 + 2m - 5m - 2m + 6 = -(m^2 - 2m - 5m + 10)

 

Group like terms: m^2 - 5m + 6 = -m^2 +7m - 10

 

Collect all terms on the left, setting the equation equal to zero: 2m^2 - 12m + 16 = 0

 

Factor: 2(m^2 - 6m + 8) = 0

           2(m - 4)(m - 2)=0

 

Set each factor equal to zero and solve: m - 4 = 0, so m = 4 and m - 2 = 0, so m = 2

 

The solutions are m = 2, 4.

 

 

Answer 2

Yikes! This leads to a cubic equation with no "simple" roots.  It will help to have a graphing calculator to solve this.  In any case, multiply both sides of the equation by the product (m²-8m+15)(m²-10m+25) to obtain:

2(m+2) = -(m-2)(m²-10m+25)  -->  2m+4 = -m³+12m²-45m+50 -->  m³-12m²+47m-46 = 0

 

The easy way to find the factors is to use the PlySmlt2 app (polynomial root finder) on a TI 84 type calculator:

 

approximate roots are 1.451696448, (5.274151776 + 1.967330233i(, and (5.274151776 - 1.967330233i)

 

You could also find the real root by graphing the function then use the CALC function to find the zero: use lower bound 0, upper bound 3, guess 1)

 

 Oops!  I wrote the solution to the product of the two fractions rather that the difference of the first two fractions

 

Here's the correct solution:

Factor the denominator expressions:
m²-8m+15 = (m-3)(m-5)
m²-10m+25 = (m-5)²      (square trinomial)
 
Now multiply each fraction in the original equation by the least common multiple, which is (m-3)(m-5)² to get:
(m+2)(m-5) - 2(m-3) = -(m-2)(m-5)  -->  m² - 3m -10 -2m + 6 = -m² + 7m - 10  --> 2m² - 12m + 6 = 0
-->  m² - 6m +3 = 0  -->  m = (6 +/- SQRT(36-4(3))/2 --> 3 +/- SQRT(6)
 
For this problem m cannot equal 3 or 5, so neither solution is extraneous