Solve the equation: `2((x)/(x+5))^2 + 5((x)/(x+5)) - 12 = 0` Hint use u = x/(x+5) and solve for x. Enter solutions separated by commas.

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tjbrewer | Elementary School Teacher | (Level 2) Associate Educator

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If we substitute using the hint `u=x/(x+5)` we come up with the equation: `2u^2+5u-12=0` This is a quadratic equation, it can easily be solved using the quadratic formula `(-b+-sqrt(b^2-4ac))/(2a)` In our case a=2, b=5 and c=-12.  We plug those values into the quadratic formula and solve

`(-5+-sqrt(5^2-4(2)(-12)))/((2)2)=(-5+-sqrt(25+96))/4=`

`(-5+-sqrt121)/4=(-5+-11)/4` That gives us `u=(-5+11)/4=6/4=1 1/2` or `u=(-5-11)/4=-16/4=-4` So u=1.5 or -4. 

Now we use these established values of u to solve for x in `x/(x+5)`

For u=1.5 `x/(x+5)=1 1/2` We multiply both sides by x+5,

`x=1 1/2(x+5)` so `x=1 1/2x+7 1/2` we subtract `1 1/2x` from each side, and we get `-1/2x=15/2` `` we multiply each side by -2 and we have `x=-15`

for u=-4 `x/(x+5)=-4` we multiply both sides by x+5,
`x=-4(x+5)` simplify and `x=-4x-20` Add 4x to both sides `5x=-20` divide each side by 5 and `x=-4`

So our final values are x=-15, -4. 

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