Find all real solutions of the equation. \frac{1}{x}+\frac{2}{x-1}=3

Clifland

Clifland

Answered question

2021-09-23

Find all real solutions of the equation.
1x+2x1=3

Answer & Explanation

StrycharzT

StrycharzT

Skilled2021-09-24Added 102 answers

Step 1
Given:
The equation 1x+2x1=3
Step 2
To solve the rational equation, multiply both sides by the LCD(Least common denominator)
Here the LCD is x(x−1)
x(x1)x+2x(x1)(x1)=3x(x1)
Simplifying, we get
x1+2x=3x23x
Move all terms to one side and make another side zero
3x23xx+12x=0
Combine the like terms
3x26x+1=0
Step 3
Use the quadratic formula to solve the quadratic equation
x=(6)±364(3)(1)2(3)
=6±266
=3±63
So, x=3+63,363
Substitute these values in the original equation and verify
Put x=3+63 in the equation 1x+2x1=3
13+63+23+631=3
Simplifying, we get both sides equal, so x=3+63 is the solution of the given equation.
Step 4
Verify the second value of x
Put x=363 in the equation 1x+2x1=3
1363+23631=3
Simplifying and rationalizing, we get both sides equal.
Therefore,
The required real solutions of the given equation are: x=3+63,363

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