Solve x+1/3x-1=1-2x+1/4

$$\frac { x + 1 } { 3 x - 1 } = 1 - \frac { 2 x + 1 } { 4 }$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{119}i}{12}\approx 0.583333333+0.909059343i$$

$$x=\frac{-\sqrt{119}i+7}{12}\approx 0.583333333-0.909059343i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 4\left(3x-1\right), the least common multiple of 3x-1,4.

$$4\left(x+1\right)=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by x+1.

$$4x+4=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by 3x-1.

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 3x-1 by 2x+1 and combine like terms.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

To find the opposite of 6x^{2}+x-1, find the opposite of each term.

$$4x+4=12x-4-6x^{2}-x+1$$

Combine 12x and -x to get 11x.

$$4x+4=11x-4-6x^{2}+1$$

Add -4 and 1 to get -3.

$$4x+4=11x-3-6x^{2}$$

Subtract 11x from both sides.

$$4x+4-11x=-3-6x^{2}$$

Combine 4x and -11x to get -7x.

$$-7x+4=-3-6x^{2}$$

Subtract -3 from both sides.

$$-7x+4-\left(-3\right)=-6x^{2}$$

The opposite of -3 is 3.

$$-7x+4+3=-6x^{2}$$

Add 6x^{2} to both sides.

$$-7x+4+3+6x^{2}=0$$

Add 4 and 3 to get 7.

$$-7x+7+6x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}-7x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -7 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\times 7}}{2\times 6}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\times 7}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 7}}{2\times 6}$$

Multiply -24 times 7.

$$x=\frac{-\left(-7\right)±\sqrt{49-168}}{2\times 6}$$

Add 49 to -168.

$$x=\frac{-\left(-7\right)±\sqrt{-119}}{2\times 6}$$

Take the square root of -119.

$$x=\frac{-\left(-7\right)±\sqrt{119}i}{2\times 6}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{119}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{7±\sqrt{119}i}{12}$$

Now solve the equation x=\frac{7±\sqrt{119}i}{12} when ± is plus. Add 7 to i\sqrt{119}.

$$x=\frac{7+\sqrt{119}i}{12}$$

Now solve the equation x=\frac{7±\sqrt{119}i}{12} when ± is minus. Subtract i\sqrt{119} from 7.

$$x=\frac{-\sqrt{119}i+7}{12}$$

The equation is now solved.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by 4\left(3x-1\right), the least common multiple of 3x-1,4.

$$4\left(x+1\right)=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by x+1.

$$4x+4=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by 3x-1.

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 3x-1 by 2x+1 and combine like terms.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

To find the opposite of 6x^{2}+x-1, find the opposite of each term.

$$4x+4=12x-4-6x^{2}-x+1$$

Combine 12x and -x to get 11x.

$$4x+4=11x-4-6x^{2}+1$$

Add -4 and 1 to get -3.

$$4x+4=11x-3-6x^{2}$$

Subtract 11x from both sides.

$$4x+4-11x=-3-6x^{2}$$

Combine 4x and -11x to get -7x.

$$-7x+4=-3-6x^{2}$$

Add 6x^{2} to both sides.

$$-7x+4+6x^{2}=-3$$

Subtract 4 from both sides.

$$-7x+6x^{2}=-3-4$$

Subtract 4 from -3 to get -7.

$$-7x+6x^{2}=-7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}-7x=-7$$

Divide both sides by 6.

$$\frac{6x^{2}-7x}{6}=-\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{7}{6}x=-\frac{7}{6}$$

Divide -\frac{7}{6}, the coefficient of the x term, by 2 to get -\frac{7}{12}. Then add the square of -\frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-\frac{7}{6}+\left(-\frac{7}{12}\right)^{2}$$

Square -\frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{7}{6}+\frac{49}{144}$$

Add -\frac{7}{6} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{119}{144}$$

Factor x^{2}-\frac{7}{6}x+\frac{49}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{12}\right)^{2}=-\frac{119}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{-\frac{119}{144}}$$

Simplify.

$$x-\frac{7}{12}=\frac{\sqrt{119}i}{12} x-\frac{7}{12}=-\frac{\sqrt{119}i}{12}$$

Add \frac{7}{12} to both sides of the equation.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$