Solve 1/3(9-42/x)=4/x-6

$$\frac { 1 } { 3 } ( 9 - \frac { 42 } { x } ) = \frac { 4 } { x } - 6$$

Solve for x

$$x=2$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of 3,x.

$$\frac{1}{3}\left(9-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9 times \frac{x}{x}.

$$\frac{1}{3}\left(\frac{9x}{x}-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

Since \frac{9x}{x} and \frac{42}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{9x-42}{x}\times 3x=3\times 4+3x\left(-6\right)$$

Cancel out 3 and 3.

$$\frac{9x-42}{x}x=3\times 4+3x\left(-6\right)$$

Express \frac{9x-42}{x}x as a single fraction.

$$\frac{\left(9x-42\right)x}{x}=3\times 4+3x\left(-6\right)$$

Multiply 3 and 4 to get 12.

$$\frac{\left(9x-42\right)x}{x}=12+3x\left(-6\right)$$

Multiply 3 and -6 to get -18.

$$\frac{\left(9x-42\right)x}{x}=12-18x$$

Use the distributive property to multiply 9x-42 by x.

$$\frac{9x^{2}-42x}{x}=12-18x$$

Add 18x to both sides.

$$\frac{9x^{2}-42x}{x}+18x=12$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18x times \frac{x}{x}.

$$\frac{9x^{2}-42x}{x}+\frac{18xx}{x}=12$$

Since \frac{9x^{2}-42x}{x} and \frac{18xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{9x^{2}-42x+18xx}{x}=12$$

Do the multiplications in 9x^{2}-42x+18xx.

$$\frac{9x^{2}-42x+18x^{2}}{x}=12$$

Combine like terms in 9x^{2}-42x+18x^{2}.

$$\frac{27x^{2}-42x}{x}=12$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$27x^{2}-42x=12x$$

Subtract 12x from both sides.

$$27x^{2}-42x-12x=0$$

Combine -42x and -12x to get -54x.

$$27x^{2}-54x=0$$

Factor out x.

$$x\left(27x-54\right)=0$$

To find equation solutions, solve x=0 and 27x-54=0.

$$x=0 x=2$$

Variable x cannot be equal to 0.

$$x=2$$

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 3x, the least common multiple of 3,x.

$$\frac{1}{3}\left(9-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9 times \frac{x}{x}.

$$\frac{1}{3}\left(\frac{9x}{x}-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

Since \frac{9x}{x} and \frac{42}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{9x-42}{x}\times 3x=3\times 4+3x\left(-6\right)$$

Cancel out 3 and 3.

$$\frac{9x-42}{x}x=3\times 4+3x\left(-6\right)$$

Express \frac{9x-42}{x}x as a single fraction.

$$\frac{\left(9x-42\right)x}{x}=3\times 4+3x\left(-6\right)$$

Multiply 3 and 4 to get 12.

$$\frac{\left(9x-42\right)x}{x}=12+3x\left(-6\right)$$

Multiply 3 and -6 to get -18.

$$\frac{\left(9x-42\right)x}{x}=12-18x$$

Use the distributive property to multiply 9x-42 by x.

$$\frac{9x^{2}-42x}{x}=12-18x$$

Subtract 12 from both sides.

$$\frac{9x^{2}-42x}{x}-12=-18x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12 times \frac{x}{x}.

$$\frac{9x^{2}-42x}{x}-\frac{12x}{x}=-18x$$

Since \frac{9x^{2}-42x}{x} and \frac{12x}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9x^{2}-42x-12x}{x}=-18x$$

Combine like terms in 9x^{2}-42x-12x.

$$\frac{9x^{2}-54x}{x}=-18x$$

Add 18x to both sides.

$$\frac{9x^{2}-54x}{x}+18x=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18x times \frac{x}{x}.

$$\frac{9x^{2}-54x}{x}+\frac{18xx}{x}=0$$

Since \frac{9x^{2}-54x}{x} and \frac{18xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{9x^{2}-54x+18xx}{x}=0$$

Do the multiplications in 9x^{2}-54x+18xx.

$$\frac{9x^{2}-54x+18x^{2}}{x}=0$$

Combine like terms in 9x^{2}-54x+18x^{2}.

$$\frac{27x^{2}-54x}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$27x^{2}-54x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -54 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}}}{2\times 27}$$

Take the square root of \left(-54\right)^{2}.

$$x=\frac{-\left(-54\right)±54}{2\times 27}$$

The opposite of -54 is 54.

$$x=\frac{54±54}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{54±54}{54}$$

Now solve the equation x=\frac{54±54}{54} when ± is plus. Add 54 to 54.

$$x=\frac{108}{54}$$

Divide 108 by 54.

$$x=2$$

Now solve the equation x=\frac{54±54}{54} when ± is minus. Subtract 54 from 54.

$$x=\frac{0}{54}$$

Divide 0 by 54.

$$x=0$$

The equation is now solved.

$$x=2 x=0$$

Variable x cannot be equal to 0.

$$x=2$$

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 3x, the least common multiple of 3,x.

$$\frac{1}{3}\left(9-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9 times \frac{x}{x}.

$$\frac{1}{3}\left(\frac{9x}{x}-\frac{42}{x}\right)\times 3x=3\times 4+3x\left(-6\right)$$

Since \frac{9x}{x} and \frac{42}{x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{3}\times \frac{9x-42}{x}\times 3x=3\times 4+3x\left(-6\right)$$

Cancel out 3 and 3.

$$\frac{9x-42}{x}x=3\times 4+3x\left(-6\right)$$

Express \frac{9x-42}{x}x as a single fraction.

$$\frac{\left(9x-42\right)x}{x}=3\times 4+3x\left(-6\right)$$

Multiply 3 and 4 to get 12.

$$\frac{\left(9x-42\right)x}{x}=12+3x\left(-6\right)$$

Multiply 3 and -6 to get -18.

$$\frac{\left(9x-42\right)x}{x}=12-18x$$

Use the distributive property to multiply 9x-42 by x.

$$\frac{9x^{2}-42x}{x}=12-18x$$

Add 18x to both sides.

$$\frac{9x^{2}-42x}{x}+18x=12$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 18x times \frac{x}{x}.

$$\frac{9x^{2}-42x}{x}+\frac{18xx}{x}=12$$

Since \frac{9x^{2}-42x}{x} and \frac{18xx}{x} have the same denominator, add them by adding their numerators.

$$\frac{9x^{2}-42x+18xx}{x}=12$$

Do the multiplications in 9x^{2}-42x+18xx.

$$\frac{9x^{2}-42x+18x^{2}}{x}=12$$

Combine like terms in 9x^{2}-42x+18x^{2}.

$$\frac{27x^{2}-42x}{x}=12$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$27x^{2}-42x=12x$$

Subtract 12x from both sides.

$$27x^{2}-42x-12x=0$$

Combine -42x and -12x to get -54x.

$$27x^{2}-54x=0$$

Divide both sides by 27.

$$\frac{27x^{2}-54x}{27}=\frac{0}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$x^{2}+\left(-\frac{54}{27}\right)x=\frac{0}{27}$$

Divide -54 by 27.

$$x^{2}-2x=\frac{0}{27}$$

Divide 0 by 27.

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

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.

$$x^{2}-2x+1=1$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$

Variable x cannot be equal to 0.

$$x=2$$