Solve 2=3/x-2-x-13/4x-2

$$2 = \frac { 3 } { x - 2 } - \frac { x - 13 } { 4 x - 2 }$$

Solve for x

$$x = \frac{\sqrt{769} + 47}{18} \approx 4.151713847$$

$$x = \frac{47 - \sqrt{769}}{18} \approx 1.070508375$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{1}{2},2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(2x-1\right), the least common multiple of x-2,4x-2.

$$4\left(x-2\right)\left(2x-1\right)=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4 by x-2.

$$\left(4x-8\right)\left(2x-1\right)=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4x-8 by 2x-1 and combine like terms.

$$8x^{2}-20x+8=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4x-2 by 3.

$$8x^{2}-20x+8=12x-6-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply x-2 by x-13 and combine like terms.

$$8x^{2}-20x+8=12x-6-\left(x^{2}-15x+26\right)$$

To find the opposite of x^{2}-15x+26, find the opposite of each term.

$$8x^{2}-20x+8=12x-6-x^{2}+15x-26$$

Combine 12x and 15x to get 27x.

$$8x^{2}-20x+8=27x-6-x^{2}-26$$

Subtract 26 from -6 to get -32.

$$8x^{2}-20x+8=27x-32-x^{2}$$

Subtract 27x from both sides.

$$8x^{2}-20x+8-27x=-32-x^{2}$$

Combine -20x and -27x to get -47x.

$$8x^{2}-47x+8=-32-x^{2}$$

Subtract -32 from both sides.

$$8x^{2}-47x+8-\left(-32\right)=-x^{2}$$

The opposite of -32 is 32.

$$8x^{2}-47x+8+32=-x^{2}$$

Add x^{2} to both sides.

$$8x^{2}-47x+8+32+x^{2}=0$$

Add 8 and 32 to get 40.

$$8x^{2}-47x+40+x^{2}=0$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}-47x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -47 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-47\right)±\sqrt{\left(-47\right)^{2}-4\times 9\times 40}}{2\times 9}$$

Square -47.

$$x=\frac{-\left(-47\right)±\sqrt{2209-4\times 9\times 40}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-47\right)±\sqrt{2209-36\times 40}}{2\times 9}$$

Multiply -36 times 40.

$$x=\frac{-\left(-47\right)±\sqrt{2209-1440}}{2\times 9}$$

Add 2209 to -1440.

$$x=\frac{-\left(-47\right)±\sqrt{769}}{2\times 9}$$

The opposite of -47 is 47.

$$x=\frac{47±\sqrt{769}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{47±\sqrt{769}}{18}$$

Now solve the equation x=\frac{47±\sqrt{769}}{18} when ± is plus. Add 47 to \sqrt{769}.

$$x=\frac{\sqrt{769}+47}{18}$$

Now solve the equation x=\frac{47±\sqrt{769}}{18} when ± is minus. Subtract \sqrt{769} from 47.

$$x=\frac{47-\sqrt{769}}{18}$$

The equation is now solved.

$$x=\frac{\sqrt{769}+47}{18} x=\frac{47-\sqrt{769}}{18}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{1}{2},2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(2x-1\right), the least common multiple of x-2,4x-2.

$$4\left(x-2\right)\left(2x-1\right)=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4 by x-2.

$$\left(4x-8\right)\left(2x-1\right)=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4x-8 by 2x-1 and combine like terms.

$$8x^{2}-20x+8=\left(4x-2\right)\times 3-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply 4x-2 by 3.

$$8x^{2}-20x+8=12x-6-\left(x-2\right)\left(x-13\right)$$

Use the distributive property to multiply x-2 by x-13 and combine like terms.

$$8x^{2}-20x+8=12x-6-\left(x^{2}-15x+26\right)$$

To find the opposite of x^{2}-15x+26, find the opposite of each term.

$$8x^{2}-20x+8=12x-6-x^{2}+15x-26$$

Combine 12x and 15x to get 27x.

$$8x^{2}-20x+8=27x-6-x^{2}-26$$

Subtract 26 from -6 to get -32.

$$8x^{2}-20x+8=27x-32-x^{2}$$

Subtract 27x from both sides.

$$8x^{2}-20x+8-27x=-32-x^{2}$$

Combine -20x and -27x to get -47x.

$$8x^{2}-47x+8=-32-x^{2}$$

Add x^{2} to both sides.

$$8x^{2}-47x+8+x^{2}=-32$$

Combine 8x^{2} and x^{2} to get 9x^{2}.

$$9x^{2}-47x+8=-32$$

Subtract 8 from both sides.

$$9x^{2}-47x=-32-8$$

Subtract 8 from -32 to get -40.

$$9x^{2}-47x=-40$$

Divide both sides by 9.

$$\frac{9x^{2}-47x}{9}=-\frac{40}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{47}{9}x=-\frac{40}{9}$$

Divide -\frac{47}{9}, the coefficient of the x term, by 2 to get -\frac{47}{18}. Then add the square of -\frac{47}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{47}{9}x+\left(-\frac{47}{18}\right)^{2}=-\frac{40}{9}+\left(-\frac{47}{18}\right)^{2}$$

Square -\frac{47}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{47}{9}x+\frac{2209}{324}=-\frac{40}{9}+\frac{2209}{324}$$

Add -\frac{40}{9} to \frac{2209}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{47}{9}x+\frac{2209}{324}=\frac{769}{324}$$

Factor x^{2}-\frac{47}{9}x+\frac{2209}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{47}{18}\right)^{2}=\frac{769}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{47}{18}\right)^{2}}=\sqrt{\frac{769}{324}}$$

Simplify.

$$x-\frac{47}{18}=\frac{\sqrt{769}}{18} x-\frac{47}{18}=-\frac{\sqrt{769}}{18}$$

Add \frac{47}{18} to both sides of the equation.

$$x=\frac{\sqrt{769}+47}{18} x=\frac{47-\sqrt{769}}{18}$$