Solve x/5-2/3x-6=4/x-2

$$\frac { x } { 5 } - \frac { 2 } { 3 x - 6 } = \frac { 4 } { x - 2 }$$

Solve for x

$$x=\frac{\sqrt{219}}{3}+1\approx 5.932882862$$

$$x=-\frac{\sqrt{219}}{3}+1\approx -3.932882862$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 15\left(x-2\right), the least common multiple of 5,3x-6,x-2.

$$\left(3x-6\right)x-5\times 2=15\times 4$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x-5\times 2=15\times 4$$

Multiply -5 and 2 to get -10.

$$3x^{2}-6x-10=15\times 4$$

Multiply 15 and 4 to get 60.

$$3x^{2}-6x-10=60$$

Subtract 60 from both sides.

$$3x^{2}-6x-10-60=0$$

Subtract 60 from -10 to get -70.

$$3x^{2}-6x-70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and -70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-70\right)}}{2\times 3}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-70\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\left(-70\right)}}{2\times 3}$$

Multiply -12 times -70.

$$x=\frac{-\left(-6\right)±\sqrt{36+840}}{2\times 3}$$

Add 36 to 840.

$$x=\frac{-\left(-6\right)±\sqrt{876}}{2\times 3}$$

Take the square root of 876.

$$x=\frac{-\left(-6\right)±2\sqrt{219}}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{219}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±2\sqrt{219}}{6}$$

Now solve the equation x=\frac{6±2\sqrt{219}}{6} when ± is plus. Add 6 to 2\sqrt{219}.

$$x=\frac{2\sqrt{219}+6}{6}$$

Divide 6+2\sqrt{219} by 6.

$$x=\frac{\sqrt{219}}{3}+1$$

Now solve the equation x=\frac{6±2\sqrt{219}}{6} when ± is minus. Subtract 2\sqrt{219} from 6.

$$x=\frac{6-2\sqrt{219}}{6}$$

Divide 6-2\sqrt{219} by 6.

$$x=-\frac{\sqrt{219}}{3}+1$$

The equation is now solved.

$$x=\frac{\sqrt{219}}{3}+1 x=-\frac{\sqrt{219}}{3}+1$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 15\left(x-2\right), the least common multiple of 5,3x-6,x-2.

$$\left(3x-6\right)x-5\times 2=15\times 4$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x-5\times 2=15\times 4$$

Multiply -5 and 2 to get -10.

$$3x^{2}-6x-10=15\times 4$$

Multiply 15 and 4 to get 60.

$$3x^{2}-6x-10=60$$

Add 10 to both sides.

$$3x^{2}-6x=60+10$$

Add 60 and 10 to get 70.

$$3x^{2}-6x=70$$

Divide both sides by 3.

$$\frac{3x^{2}-6x}{3}=\frac{70}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{6}{3}\right)x=\frac{70}{3}$$

Divide -6 by 3.

$$x^{2}-2x=\frac{70}{3}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{70}{3}+1$$

Add \frac{70}{3} to 1.

$$x^{2}-2x+1=\frac{73}{3}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{73}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{73}{3}}$$

Simplify.

$$x-1=\frac{\sqrt{219}}{3} x-1=-\frac{\sqrt{219}}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{219}}{3}+1 x=-\frac{\sqrt{219}}{3}+1$$