Solve for x (4x)/(x^2+x+3)+(5x)/(x^2-5x+3)=-3/2

Problem

(4*x)/(x2+x+3)+(5*x)/(x2−5*x+3)=−3/2

Solution

1. Divide both sides of the equation by x (noting that x=0 is not a solution) to simplify the fractions.

4/(x+1+3/x)+5/(x−5+3/x)=−3/(2*x)

2. Substitute a new variable y=x+3/x to reduce the complexity of the denominators.

4/(y+1)+5/(y−5)=−3/2

3. Find a common denominator for the left side of the equation.

(4*(y−5)+5*(y+1))/((y+1)*(y−5))=−3/2

4. Simplify the numerator and denominator.

(9*y−15)/(y2−4*y−5)=−3/2

5. Cross-multiply to eliminate the fractions.

2*(9*y−15)=−3*(y2−4*y−5)

6. Expand and move all terms to one side to form a quadratic equation.

18*y−30=−3*y2+12*y+15

3*y2+6*y−45=0

7. Divide the equation by 3 and factor the resulting quadratic.

y2+2*y−15=0

(y+5)*(y−3)=0

8. Solve for y giving two possible values.

y=−5

y=3

9. Substitute back y=x+3/x for the first case y=−5

x+3/x=−5

x2+5*x+3=0

10. Apply the quadratic formula to find the first set of solutions for x

x=(−5±√(,25−12))/2

x=(−5±√(,13))/2

11. Substitute back y=x+3/x for the second case y=3

x+3/x=3

x2−3*x+3=0

12. Check the discriminant for the second quadratic equation.

D=(−3)2−4*(1)*(3)=9−12=−3

Since the discriminant is negative, there are no real solutions for this case.

Final Answer

(4*x)/(x2+x+3)+(5*x)/(x2−5*x+3)=−3/2⇒x=(−5±√(,13))/2

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