Solve x-1/(2x)=3+1/x

Question

Solve the equation

x_{1} = \frac{3 - 15}{2}, x_{2} = \frac{3 + 15}{2}

Alternative Form

x_{1} \approx - 0.436492, x_{2} \approx 3.436492

Evaluate

x - \frac{1}{2 x} = 3 + \frac{1}{x}

Find the domain

More Steps

x - \frac{1}{2 x} = 3 + \frac{1}{x}, x \neq = 0

Multiply both sides of the equation by LCD

(x - \frac{1}{2 x}) \times 2 x = (3 + \frac{1}{x}) \times 2 x

Simplify the equation

More Steps

2 x^{2} - 1 = (3 + \frac{1}{x}) \times 2 x

Simplify the equation

More Steps

2 x^{2} - 1 = 6 x + 2

Move the expression to the left side

2 x^{2} - 1 - (6 x + 2) = 0

Subtract the terms

More Steps

2 x^{2} - 3 - 6 x = 0

Rewrite in standard form

2 x^{2} - 6 x - 3 = 0

Substitute a= 2,b= - 6 and c= - 3 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{6 \pm ( - 6 ) ^{2} - 4 \times 2 ( - 3 )}{2 \times 2}

Simplify the expression

x = \frac{6 \pm ( - 6 ) ^{2} - 4 \times 2 ( - 3 )}{4}

Simplify the expression

More Steps

x = \frac{6 \pm 60}{4}

Simplify the radical expression

More Steps

x = \frac{6 \pm 2 15}{4}

Separate the equation into 2 possible cases

x = \frac{6 + 2 15}{4} x = \frac{6 - 2 15}{4}

Simplify the expression

More Steps

x = \frac{3 + 15}{2} x = \frac{6 - 2 15}{4}

Simplify the expression

More Steps

x = \frac{3 + 15}{2} x = \frac{3 - 15}{2}

Check if the solution is in the defined range

x = \frac{3 + 15}{2} x = \frac{3 - 15}{2}, x \neq = 0

Find the intersection of the solution and the defined range

x = \frac{3 + 15}{2} x = \frac{3 - 15}{2}

Solution

x_{1} = \frac{3 - 15}{2}, x_{2} = \frac{3 + 15}{2}

Alternative Form

x_{1} \approx - 0.436492, x_{2} \approx 3.436492

Show Solution