Solve 4/x-2-x-3/x-4=0

$$\frac{ 4 }{ x-2 } - \frac{ x-3 }{ x-4 } =0$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{7}i+9}{2}\approx 4.5-1.322875656i$$

$$x=\frac{9+\sqrt{7}i}{2}\approx 4.5+1.322875656i$$

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.

$$\left(x-4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x^{2}-5x+6\right)=0$$

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

$$4x-16-x^{2}+5x-6=0$$

Combine 4x and 5x to get 9x.

$$9x-16-x^{2}-6=0$$

Subtract 6 from -16 to get -22.

$$9x-22-x^{2}=0$$

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.

$$-x^{2}+9x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\left(-22\right)}}{2\left(-1\right)}$$

Multiply 4 times -22.

$$x=\frac{-9±\sqrt{81-88}}{2\left(-1\right)}$$

Add 81 to -88.

$$x=\frac{-9±\sqrt{-7}}{2\left(-1\right)}$$

Take the square root of -7.

$$x=\frac{-9±\sqrt{7}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±\sqrt{7}i}{-2}$$

Now solve the equation x=\frac{-9±\sqrt{7}i}{-2} when ± is plus. Add -9 to i\sqrt{7}.

$$x=\frac{-9+\sqrt{7}i}{-2}$$

Divide -9+i\sqrt{7} by -2.

$$x=\frac{-\sqrt{7}i+9}{2}$$

Now solve the equation x=\frac{-9±\sqrt{7}i}{-2} when ± is minus. Subtract i\sqrt{7} from -9.

$$x=\frac{-\sqrt{7}i-9}{-2}$$

Divide -9-i\sqrt{7} by -2.

$$x=\frac{9+\sqrt{7}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{7}i+9}{2} x=\frac{9+\sqrt{7}i}{2}$$

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.

$$\left(x-4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x^{2}-5x+6\right)=0$$

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

$$4x-16-x^{2}+5x-6=0$$

Combine 4x and 5x to get 9x.

$$9x-16-x^{2}-6=0$$

Subtract 6 from -16 to get -22.

$$9x-22-x^{2}=0$$

Add 22 to both sides. Anything plus zero gives itself.

$$9x-x^{2}=22$$

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.

$$-x^{2}+9x=22$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=\frac{22}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=\frac{22}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=\frac{22}{-1}$$

Divide 22 by -1.

$$x^{2}-9x=-22$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-22+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-22+\frac{81}{4}$$

Add -22 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=-\frac{7}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{7}i}{2} x-\frac{9}{2}=-\frac{\sqrt{7}i}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{9+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+9}{2}$$