Solve 1/(3x-4)=2(x+6)

$$1 / ( 3 x - 4 ) = 2 ( x + 6 )$$

Solve for x

$$x=\frac{7\sqrt{10}}{6}-\frac{7}{3}\approx 1.355990604$$

$$x=-\frac{7\sqrt{10}}{6}-\frac{7}{3}\approx -6.02265727$$

Steps Using the Quadratic Formula

Variable x cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 3x-4.

$$1=2\left(x+6\right)\left(3x-4\right)$$

Use the distributive property to multiply 2 by x+6.

$$1=\left(2x+12\right)\left(3x-4\right)$$

Use the distributive property to multiply 2x+12 by 3x-4 and combine like terms.

$$1=6x^{2}+28x-48$$

Swap sides so that all variable terms are on the left hand side.

$$6x^{2}+28x-48=1$$

Subtract 1 from both sides.

$$6x^{2}+28x-48-1=0$$

Subtract 1 from -48 to get -49.

$$6x^{2}+28x-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 28 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\times 6\left(-49\right)}}{2\times 6}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 6\left(-49\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-28±\sqrt{784-24\left(-49\right)}}{2\times 6}$$

Multiply -24 times -49.

$$x=\frac{-28±\sqrt{784+1176}}{2\times 6}$$

Add 784 to 1176.

$$x=\frac{-28±\sqrt{1960}}{2\times 6}$$

Take the square root of 1960.

$$x=\frac{-28±14\sqrt{10}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-28±14\sqrt{10}}{12}$$

Now solve the equation x=\frac{-28±14\sqrt{10}}{12} when ± is plus. Add -28 to 14\sqrt{10}.

$$x=\frac{14\sqrt{10}-28}{12}$$

Divide -28+14\sqrt{10} by 12.

$$x=\frac{7\sqrt{10}}{6}-\frac{7}{3}$$

Now solve the equation x=\frac{-28±14\sqrt{10}}{12} when ± is minus. Subtract 14\sqrt{10} from -28.

$$x=\frac{-14\sqrt{10}-28}{12}$$

Divide -28-14\sqrt{10} by 12.

$$x=-\frac{7\sqrt{10}}{6}-\frac{7}{3}$$

The equation is now solved.

$$x=\frac{7\sqrt{10}}{6}-\frac{7}{3} x=-\frac{7\sqrt{10}}{6}-\frac{7}{3}$$

Steps for Completing the Square

Variable x cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 3x-4.

$$1=2\left(x+6\right)\left(3x-4\right)$$

Use the distributive property to multiply 2 by x+6.

$$1=\left(2x+12\right)\left(3x-4\right)$$

Use the distributive property to multiply 2x+12 by 3x-4 and combine like terms.

$$1=6x^{2}+28x-48$$

Swap sides so that all variable terms are on the left hand side.

$$6x^{2}+28x-48=1$$

Add 48 to both sides.

$$6x^{2}+28x=1+48$$

Add 1 and 48 to get 49.

$$6x^{2}+28x=49$$

Divide both sides by 6.

$$\frac{6x^{2}+28x}{6}=\frac{49}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{28}{6}x=\frac{49}{6}$$

Reduce the fraction \frac{28}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{14}{3}x=\frac{49}{6}$$

Divide \frac{14}{3}, the coefficient of the x term, by 2 to get \frac{7}{3}. Then add the square of \frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{14}{3}x+\left(\frac{7}{3}\right)^{2}=\frac{49}{6}+\left(\frac{7}{3}\right)^{2}$$

Square \frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=\frac{49}{6}+\frac{49}{9}$$

Add \frac{49}{6} to \frac{49}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=\frac{245}{18}$$

Factor x^{2}+\frac{14}{3}x+\frac{49}{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}.

$$\left(x+\frac{7}{3}\right)^{2}=\frac{245}{18}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{3}\right)^{2}}=\sqrt{\frac{245}{18}}$$

Simplify.

$$x+\frac{7}{3}=\frac{7\sqrt{10}}{6} x+\frac{7}{3}=-\frac{7\sqrt{10}}{6}$$

Subtract \frac{7}{3} from both sides of the equation.

$$x=\frac{7\sqrt{10}}{6}-\frac{7}{3} x=-\frac{7\sqrt{10}}{6}-\frac{7}{3}$$