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

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

Solve for x

$$x=\frac{\sqrt{3}}{3}+3\approx 3.577350269$$

$$x=-\frac{\sqrt{3}}{3}+3\approx 2.422649731$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 2,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-2\right), the least common multiple of x-2,x-3,x-4.

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

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x^{2}-7x+12+\left(x-4\right)\left(x-2\right)=-\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-4 by x-2 and combine like terms.

$$x^{2}-7x+12+x^{2}-6x+8=-\left(x-3\right)\left(x-2\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-7x+12-6x+8=-\left(x-3\right)\left(x-2\right)$$

Combine -7x and -6x to get -13x.

$$2x^{2}-13x+12+8=-\left(x-3\right)\left(x-2\right)$$

Add 12 and 8 to get 20.

$$2x^{2}-13x+20=-\left(x-3\right)\left(x-2\right)$$

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

$$2x^{2}-13x+20=-\left(x^{2}-5x+6\right)$$

To find the opposite of x^{2}-5x+6, find the opposite of each term.

$$2x^{2}-13x+20=-x^{2}+5x-6$$

Add x^{2} to both sides.

$$2x^{2}-13x+20+x^{2}=5x-6$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}-13x+20=5x-6$$

Subtract 5x from both sides.

$$3x^{2}-13x+20-5x=-6$$

Combine -13x and -5x to get -18x.

$$3x^{2}-18x+20=-6$$

Add 6 to both sides.

$$3x^{2}-18x+20+6=0$$

Add 20 and 6 to get 26.

$$3x^{2}-18x+26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -18 for b, and 26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\times 26}}{2\times 3}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 3\times 26}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-18\right)±\sqrt{324-12\times 26}}{2\times 3}$$

Multiply -12 times 26.

$$x=\frac{-\left(-18\right)±\sqrt{324-312}}{2\times 3}$$

Add 324 to -312.

$$x=\frac{-\left(-18\right)±\sqrt{12}}{2\times 3}$$

Take the square root of 12.

$$x=\frac{-\left(-18\right)±2\sqrt{3}}{2\times 3}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{18±2\sqrt{3}}{6}$$

Now solve the equation x=\frac{18±2\sqrt{3}}{6} when ± is plus. Add 18 to 2\sqrt{3}.

$$x=\frac{2\sqrt{3}+18}{6}$$

Divide 18+2\sqrt{3} by 6.

$$x=\frac{\sqrt{3}}{3}+3$$

Now solve the equation x=\frac{18±2\sqrt{3}}{6} when ± is minus. Subtract 2\sqrt{3} from 18.

$$x=\frac{18-2\sqrt{3}}{6}$$

Divide 18-2\sqrt{3} by 6.

$$x=-\frac{\sqrt{3}}{3}+3$$

The equation is now solved.

$$x=\frac{\sqrt{3}}{3}+3 x=-\frac{\sqrt{3}}{3}+3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 2,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-2\right), the least common multiple of x-2,x-3,x-4.

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

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x^{2}-7x+12+\left(x-4\right)\left(x-2\right)=-\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-4 by x-2 and combine like terms.

$$x^{2}-7x+12+x^{2}-6x+8=-\left(x-3\right)\left(x-2\right)$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-7x+12-6x+8=-\left(x-3\right)\left(x-2\right)$$

Combine -7x and -6x to get -13x.

$$2x^{2}-13x+12+8=-\left(x-3\right)\left(x-2\right)$$

Add 12 and 8 to get 20.

$$2x^{2}-13x+20=-\left(x-3\right)\left(x-2\right)$$

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

$$2x^{2}-13x+20=-\left(x^{2}-5x+6\right)$$

To find the opposite of x^{2}-5x+6, find the opposite of each term.

$$2x^{2}-13x+20=-x^{2}+5x-6$$

Add x^{2} to both sides.

$$2x^{2}-13x+20+x^{2}=5x-6$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}-13x+20=5x-6$$

Subtract 5x from both sides.

$$3x^{2}-13x+20-5x=-6$$

Combine -13x and -5x to get -18x.

$$3x^{2}-18x+20=-6$$

Subtract 20 from both sides.

$$3x^{2}-18x=-6-20$$

Subtract 20 from -6 to get -26.

$$3x^{2}-18x=-26$$

Divide both sides by 3.

$$\frac{3x^{2}-18x}{3}=-\frac{26}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{18}{3}\right)x=-\frac{26}{3}$$

Divide -18 by 3.

$$x^{2}-6x=-\frac{26}{3}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-\frac{26}{3}+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-\frac{26}{3}+9$$

Add -\frac{26}{3} to 9.

$$x^{2}-6x+9=\frac{1}{3}$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=\frac{1}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{\frac{1}{3}}$$

Simplify.

$$x-3=\frac{\sqrt{3}}{3} x-3=-\frac{\sqrt{3}}{3}$$

Add 3 to both sides of the equation.

$$x=\frac{\sqrt{3}}{3}+3 x=-\frac{\sqrt{3}}{3}+3$$