Solve each equation

Question

5/x^2-7x+12-2/3-x =5/x-4

Answer 1

5/x^2 - 7x - 12 -2/(3-x) = 5/(x-4)

multiply by common denominator

5(3-x)(x-4) - 7x^3(3-x)(x-4) - 12x(3-x)(x-4) - 2x(x-4) = 5x(3-x)

expand, combine like terms to get a 5th degree polynomial equation

-5(x^2 -7x +12) - 7x^3(x^2 -7x +12) -12x(x^2 -7x+12) = 15x-5x^2

-5x^2 +35x -60 -7x^5 +49x^4 -84x -12x^3 +84x -144 = 15x -5x^2

-7x^5 +49x^4 -12x^3 +20x -204 = 0

factor by groping, if possible.

or

try graphing it and find the points where the curve intersects the x axis

also decartes' rule on signs narrows down the possible zeros

3 possible positive real solutions

1 possible positive real solution

as imaginary solutions come in pairs

that leaves

max 3 real (2 +, 1-)and 2 imaginary

or 1 positive real, 4 imaginary

meaning at least 1 positive real solution

x f(x)

0 -204

1 -7+49-12 +20-204 = 69 -223 = - 154

2 -7(2^5) +49(2^4) -12(2^3) +20(2) -204 = -14(16)+49(16)-6(16) +40-204 = 29(16) -164 = 464-164 = 300

3 -7(3)^5 +49(3)^4 -12(3)^3 +20(3) -204 = -21(81) + 49(81) -4(81) +60-204

= 24(81) -144 = 3400

the sign change from f(1) to f(2), - to +, means one zero between x=1 and x=2

1<x<2 likely has one solution, but theoretically it might have 3 solutions between 1 and 2

Unless you reall meant the equation as written with no implicity parentheses in two denominators

then

5/x^2 -7x +12 -2/3 -x= 2/x -4

5 -7x^3 +12x -2x^2/3 -x^3 = 2x -4x^2

-8x^3 +(10/3)x^2+10x +5 = 0

1 possible positive real solution

2 possible negative real solutions

either

1 positive real and 2 imaginary

or

1 negative real and 2 imaginary

either 1 positive real or 1 negative real

x f(x)

0 5

1 3 2/3

2 -64 +40/3 +25 = -64 + 38 1/3 = - 25 2/3

the sign change shows the only real solution is between x=1 and x=2

closer to 1, such as, a wild guess, maybe 1.2

which is roughly coincidentally the same answer as the 1st interpretation of the equation and fractions

which means order of operattions doesn't change the answer, at least not more than a small fraction.