Solve 2x-5/x-2=-x-2/x-4

$$\frac { 2 x - 5 } { x - 2 } = \frac { - x - 2 } { x - 4 }$$

Solve for x (complex solution)

$$x=\frac{13+\sqrt{23}i}{6}\approx 2.166666667+0.799305254i$$

$$x=\frac{-\sqrt{23}i+13}{6}\approx 2.166666667-0.799305254i$$

Steps Using the Quadratic Formula

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

$$\left(x-4\right)\left(2x-5\right)=\left(x-2\right)\left(-x-2\right)$$

Use the distributive property to multiply x-4 by 2x-5 and combine like terms.

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

Use the distributive property to multiply x-2 by -x-2.

$$2x^{2}-13x+20=x\left(-x\right)-2x-2\left(-x\right)+4$$

Multiply -2 and -1 to get 2.

$$2x^{2}-13x+20=x\left(-x\right)-2x+2x+4$$

Combine -2x and 2x to get 0.

$$2x^{2}-13x+20=x\left(-x\right)+4$$

Subtract x\left(-x\right) from both sides.

$$2x^{2}-13x+20-x\left(-x\right)=4$$

Subtract 4 from both sides.

$$2x^{2}-13x+20-x\left(-x\right)-4=0$$

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

$$2x^{2}-13x+20-x^{2}\left(-1\right)-4=0$$

Multiply -1 and -1 to get 1.

$$2x^{2}-13x+20+x^{2}-4=0$$

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

$$3x^{2}-13x+20-4=0$$

Subtract 4 from 20 to get 16.

$$3x^{2}-13x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -13 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\times 16}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\times 16}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\times 16}}{2\times 3}$$

Multiply -12 times 16.

$$x=\frac{-\left(-13\right)±\sqrt{169-192}}{2\times 3}$$

Add 169 to -192.

$$x=\frac{-\left(-13\right)±\sqrt{-23}}{2\times 3}$$

Take the square root of -23.

$$x=\frac{-\left(-13\right)±\sqrt{23}i}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{23}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±\sqrt{23}i}{6}$$

Now solve the equation x=\frac{13±\sqrt{23}i}{6} when ± is plus. Add 13 to i\sqrt{23}.

$$x=\frac{13+\sqrt{23}i}{6}$$

Now solve the equation x=\frac{13±\sqrt{23}i}{6} when ± is minus. Subtract i\sqrt{23} from 13.

$$x=\frac{-\sqrt{23}i+13}{6}$$

The equation is now solved.

$$x=\frac{13+\sqrt{23}i}{6} x=\frac{-\sqrt{23}i+13}{6}$$

Steps for Completing the Square

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

$$\left(x-4\right)\left(2x-5\right)=\left(x-2\right)\left(-x-2\right)$$

Use the distributive property to multiply x-4 by 2x-5 and combine like terms.

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

Use the distributive property to multiply x-2 by -x-2.

$$2x^{2}-13x+20=x\left(-x\right)-2x-2\left(-x\right)+4$$

Multiply -2 and -1 to get 2.

$$2x^{2}-13x+20=x\left(-x\right)-2x+2x+4$$

Combine -2x and 2x to get 0.

$$2x^{2}-13x+20=x\left(-x\right)+4$$

Subtract x\left(-x\right) from both sides.

$$2x^{2}-13x+20-x\left(-x\right)=4$$

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

$$2x^{2}-13x+20-x^{2}\left(-1\right)=4$$

Multiply -1 and -1 to get 1.

$$2x^{2}-13x+20+x^{2}=4$$

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

$$3x^{2}-13x+20=4$$

Subtract 20 from both sides.

$$3x^{2}-13x=4-20$$

Subtract 20 from 4 to get -16.

$$3x^{2}-13x=-16$$

Divide both sides by 3.

$$\frac{3x^{2}-13x}{3}=-\frac{16}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{13}{3}x=-\frac{16}{3}$$

Divide -\frac{13}{3}, the coefficient of the x term, by 2 to get -\frac{13}{6}. Then add the square of -\frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{3}x+\left(-\frac{13}{6}\right)^{2}=-\frac{16}{3}+\left(-\frac{13}{6}\right)^{2}$$

Square -\frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=-\frac{16}{3}+\frac{169}{36}$$

Add -\frac{16}{3} to \frac{169}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=-\frac{23}{36}$$

Factor x^{2}-\frac{13}{3}x+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{6}\right)^{2}=-\frac{23}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{6}\right)^{2}}=\sqrt{-\frac{23}{36}}$$

Simplify.

$$x-\frac{13}{6}=\frac{\sqrt{23}i}{6} x-\frac{13}{6}=-\frac{\sqrt{23}i}{6}$$

Add \frac{13}{6} to both sides of the equation.

$$x=\frac{13+\sqrt{23}i}{6} x=\frac{-\sqrt{23}i+13}{6}$$