What are the roots of the equation x / (x + 2) + 3 / (x - 4).
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What you have given is an expression. Equating it to 0, the roots have been calculated.
x / (x + 2) + 3 / (x - 4) = 0
=> x( x - 4) + 3(x + 2) = 0
=> x^2 - 4x + 6 + 3x = 0
=> x^2 - x + 6 = 0
The roots of this equation are
[ 1 + sqrt (1^2 - 24) ] /2
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Question: What are the roots of the equation x / (x + 2) + 3 / (x -4) = 0.(edited please).
A:
We multiply both sides by (x+2)(x-4):
x(x-4) + 3(x+2) = 0.
x^2-4x+3x +6 = 0.
x^2-x+6 = 0.
We know that ax^2+bx+c = 0 has roots:
x1 = {-b+(b^2-4ac)^(1/2)}/2a or x2 = {-b-(b^2-4ac)^(1/2)}/2a.
So x^2-x+3 has roots:
x1 = {-(-1)+(1-4*1*6)^(1/2)}/2 = {1+(-23)^(1/2)}/2, or
x2 = {1-(-23)^(1/2)}/2.
So the roots are x1 = {1+(-23)^(1/2)}/2 and x2 = {1-(-23)^(1/2)}/2.
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