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

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

Solve for x

$$x=\frac{5\sqrt{66}}{6}-\frac{20}{3}\approx 0.103365337$$

$$x=-\frac{5\sqrt{66}}{6}-\frac{20}{3}\approx -13.436698671$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 30x, the least common multiple of 5,6x.

$$\frac{3}{5}x\times 30x+30x\times 2=5\times 5+30x\left(-6\right)$$

Multiply \frac{3}{5} and 30 to get 18.

$$18xx+30x\times 2=5\times 5+30x\left(-6\right)$$

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

$$18x^{2}+30x\times 2=5\times 5+30x\left(-6\right)$$

Multiply 30 and 2 to get 60.

$$18x^{2}+60x=5\times 5+30x\left(-6\right)$$

Multiply 5 and 5 to get 25.

$$18x^{2}+60x=25+30x\left(-6\right)$$

Multiply 30 and -6 to get -180.

$$18x^{2}+60x=25-180x$$

Subtract 25 from both sides.

$$18x^{2}+60x-25=-180x$$

Add 180x to both sides.

$$18x^{2}+60x-25+180x=0$$

Combine 60x and 180x to get 240x.

$$18x^{2}+240x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 240 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-240±\sqrt{240^{2}-4\times 18\left(-25\right)}}{2\times 18}$$

Square 240.

$$x=\frac{-240±\sqrt{57600-4\times 18\left(-25\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-240±\sqrt{57600-72\left(-25\right)}}{2\times 18}$$

Multiply -72 times -25.

$$x=\frac{-240±\sqrt{57600+1800}}{2\times 18}$$

Add 57600 to 1800.

$$x=\frac{-240±\sqrt{59400}}{2\times 18}$$

Take the square root of 59400.

$$x=\frac{-240±30\sqrt{66}}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-240±30\sqrt{66}}{36}$$

Now solve the equation x=\frac{-240±30\sqrt{66}}{36} when ± is plus. Add -240 to 30\sqrt{66}.

$$x=\frac{30\sqrt{66}-240}{36}$$

Divide -240+30\sqrt{66} by 36.

$$x=\frac{5\sqrt{66}}{6}-\frac{20}{3}$$

Now solve the equation x=\frac{-240±30\sqrt{66}}{36} when ± is minus. Subtract 30\sqrt{66} from -240.

$$x=\frac{-30\sqrt{66}-240}{36}$$

Divide -240-30\sqrt{66} by 36.

$$x=-\frac{5\sqrt{66}}{6}-\frac{20}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{66}}{6}-\frac{20}{3} x=-\frac{5\sqrt{66}}{6}-\frac{20}{3}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 30x, the least common multiple of 5,6x.

$$\frac{3}{5}x\times 30x+30x\times 2=5\times 5+30x\left(-6\right)$$

Multiply \frac{3}{5} and 30 to get 18.

$$18xx+30x\times 2=5\times 5+30x\left(-6\right)$$

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

$$18x^{2}+30x\times 2=5\times 5+30x\left(-6\right)$$

Multiply 30 and 2 to get 60.

$$18x^{2}+60x=5\times 5+30x\left(-6\right)$$

Multiply 5 and 5 to get 25.

$$18x^{2}+60x=25+30x\left(-6\right)$$

Multiply 30 and -6 to get -180.

$$18x^{2}+60x=25-180x$$

Add 180x to both sides.

$$18x^{2}+60x+180x=25$$

Combine 60x and 180x to get 240x.

$$18x^{2}+240x=25$$

Divide both sides by 18.

$$\frac{18x^{2}+240x}{18}=\frac{25}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{240}{18}x=\frac{25}{18}$$

Reduce the fraction \frac{240}{18} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{40}{3}x=\frac{25}{18}$$

Divide \frac{40}{3}, the coefficient of the x term, by 2 to get \frac{20}{3}. Then add the square of \frac{20}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{40}{3}x+\left(\frac{20}{3}\right)^{2}=\frac{25}{18}+\left(\frac{20}{3}\right)^{2}$$

Square \frac{20}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{40}{3}x+\frac{400}{9}=\frac{25}{18}+\frac{400}{9}$$

Add \frac{25}{18} to \frac{400}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{40}{3}x+\frac{400}{9}=\frac{275}{6}$$

Factor x^{2}+\frac{40}{3}x+\frac{400}{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}.

$$\left(x+\frac{20}{3}\right)^{2}=\frac{275}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{20}{3}\right)^{2}}=\sqrt{\frac{275}{6}}$$

Simplify.

$$x+\frac{20}{3}=\frac{5\sqrt{66}}{6} x+\frac{20}{3}=-\frac{5\sqrt{66}}{6}$$

Subtract \frac{20}{3} from both sides of the equation.

$$x=\frac{5\sqrt{66}}{6}-\frac{20}{3} x=-\frac{5\sqrt{66}}{6}-\frac{20}{3}$$