Question 866191
{{{((x^2-2x+4)/((2x)(x+1)(x-3)))=A/(2x)+B/(x+1)+C/(x-3)}}}
{{{x^2-2x+4=A(x+1)(x-3)+2Bx(x-3)+2Cx(x+1)}}}
{{{x^2-2x+4=A(x^2-2x-3)+(2Bx^2-6Bx)+(2Cx^2+2Cx)}}}
{{{x^2-2x+4=(Ax^2-2Ax-3A)+(2Bx^2-6Bx)+(2Cx^2+2Cx)}}}
{{{x^2-2x+4=(A+2B+2C)x^2+(-2A-6B+2C)x+(-3A)}}}
Comparing terms,
1.{{{A+2B+2C=1}}}
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2.{{{-2A-6B+2C=-2}}}
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3.{{{-3A=4}}}
From eq. 3,
{{{A=-4/3}}}
Add eq. 1 and 2 together,
{{{A+2B+2C+2A+6B-2C=1+2}}}
{{{3A+8B=3}}}
{{{-4+8B=3}}}
{{{8B=7}}}
{{{B=7/8}}}
and finally,
{{{A+2B+2C=1}}}
{{{-4/3+14/8+2C=1}}}
{{{2C=24/24+32/24-42/24}}}
{{{2C=14/24}}}
{{{C=7/24}}}
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{{{((x^2-2x+4)/((2x)(x+1)(x-3)))=A/(2x)+B/(x+1)+C/(x-3)}}}
{{{((x^2-2x+4)/((2x)(x+1)(x-3)))=-(4/3)/(2x)+(7/8)/(x+1)+(7/24)/(x-3)}}}
{{{((x^2-2x+4)/((2x)(x+1)(x-3)))=-2/(3x)+7/(8(x+1))+7/(24(x-3))}}}
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{{{((-4y)/(3y^2-4y+1))=A/(3y-1) +B/(y-1)}}}
{{{-4y=A(y-1)+B(3y-1)}}}
{{{-4y=Ay-A+3By-B}}}
{{{-4y=(A+3B)y+(-A-B)}}}
Which leads to,
1.{{{A+3B=-4}}}
2.{{{-A-B=0}}}
From eq. 2,
{{{A=-B}}}
Substitute into eq. 1,
{{{-B+3B=-4}}}
{{{2B=-4}}}
{{{B=-2}}}
and
{{{A=2}}}
{{{((-4y)/(3y^2-4y+1))=A/(3y-1) +B/(y-1)}}}
{{{((-4y)/(3y^2-4y+1))=2/(3y-1)-2/(y-1)}}}