Solve for x x+(4x)/(x-3)=12/(x-3)

Problem

x+(4*x)/(x−3)=12/(x−3)

Solution

1. Identify the domain by noting that the denominator x−3 cannot be zero, which means x≠3

2. Eliminate the fractions by multiplying every term in the equation by the least common denominator, which is x−3

x*(x−3)+(x−3)(4*x)/(x−3)=(x−3)12/(x−3)

3. Simplify the equation by performing the multiplication and canceling the denominators.

x2−3*x+4*x=12

4. Combine like terms on the left side of the equation.

x2+x=12

5. Set the quadratic to zero by subtracting 12 from both sides to prepare for factoring.

x2+x−12=0

6. Factor the quadratic expression by finding two numbers that multiply to −12 and add to 1

(x+4)*(x−3)=0

7. Solve for x by setting each factor equal to zero.

x+4=0⇒x=−4

x−3=0⇒x=3

8. Check against the domain and observe that x=3 is an extraneous solution because it makes the original denominator zero.

Final Answer

x=−4

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