Solve 1/1frac{7{9}x}+1/x+1/1frac{7{9}x+2}=1/12

$$\frac { 1 } { 1 \frac { 7 } { 9 } x } + \frac { 1 } { x } + \frac { 1 } { 1 \frac { 7 } { 9 } x + 2 } = \frac { 1 } { 12 }$$

Solve for x

$$x = \frac{15 \sqrt{193} + 195}{16} \approx 25.21166624$$

$$x=\frac{195-15\sqrt{193}}{16}\approx -0.83666624$$

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 12x, the least common multiple of x,12.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

Add \frac{27}{4} and 12 to get \frac{75}{4}.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Subtract x from both sides.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Reorder the terms.

$$-x+54\times \frac{1}{8x+9}x+\frac{75}{4}=0$$

Variable x cannot be equal to -\frac{9}{8} since division by zero is not defined. Multiply both sides of the equation by 4\left(8x+9\right), the least common multiple of 8x+9,4.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Multiply -1 and 4 to get -4.

$$-4x\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Use the distributive property to multiply -4x by 8x+9.

$$-32x^{2}-36x+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Multiply 54 and 4 to get 216.

$$-32x^{2}-36x+216\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Multiply 216 and 1 to get 216.

$$-32x^{2}-36x+216x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Combine -36x and 216x to get 180x.

$$-32x^{2}+180x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Multiply 4 and \frac{75}{4} to get 75.

$$-32x^{2}+180x+75\left(8x+9\right)=0$$

Use the distributive property to multiply 75 by 8x+9.

$$-32x^{2}+180x+600x+675=0$$

Combine 180x and 600x to get 780x.

$$-32x^{2}+780x+675=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, 780 for b, and 675 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-780±\sqrt{780^{2}-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

Square 780.

$$x=\frac{-780±\sqrt{608400-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

Multiply -4 times -32.

$$x=\frac{-780±\sqrt{608400+128\times 675}}{2\left(-32\right)}$$

Multiply 128 times 675.

$$x=\frac{-780±\sqrt{608400+86400}}{2\left(-32\right)}$$

Add 608400 to 86400.

$$x=\frac{-780±\sqrt{694800}}{2\left(-32\right)}$$

Take the square root of 694800.

$$x=\frac{-780±60\sqrt{193}}{2\left(-32\right)}$$

Multiply 2 times -32.

$$x=\frac{-780±60\sqrt{193}}{-64}$$

Now solve the equation x=\frac{-780±60\sqrt{193}}{-64} when ± is plus. Add -780 to 60\sqrt{193}.

$$x=\frac{60\sqrt{193}-780}{-64}$$

Divide -780+60\sqrt{193} by -64.

$$x=\frac{195-15\sqrt{193}}{16}$$

Now solve the equation x=\frac{-780±60\sqrt{193}}{-64} when ± is minus. Subtract 60\sqrt{193} from -780.

$$x=\frac{-60\sqrt{193}-780}{-64}$$

Divide -780-60\sqrt{193} by -64.

$$x=\frac{15\sqrt{193}+195}{16}$$

The equation is now solved.

$$x=\frac{195-15\sqrt{193}}{16} x=\frac{15\sqrt{193}+195}{16}$$

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 12x, the least common multiple of x,12.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

Add \frac{27}{4} and 12 to get \frac{75}{4}.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Subtract x from both sides.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Subtract \frac{75}{4} from both sides. Anything subtracted from zero gives its negation.

$$54x\left(8x+9\right)^{-1}-x=-\frac{75}{4}$$

Reorder the terms.

$$-x+54\times \frac{1}{8x+9}x=-\frac{75}{4}$$

Variable x cannot be equal to -\frac{9}{8} since division by zero is not defined. Multiply both sides of the equation by 4\left(8x+9\right), the least common multiple of 8x+9,4.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

Multiply -1 and 4 to get -4.

$$-4x\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

Use the distributive property to multiply -4x by 8x+9.

$$-32x^{2}-36x+54\times 4\times 1x=-75\left(8x+9\right)$$

Multiply 54 and 4 to get 216.

$$-32x^{2}-36x+216\times 1x=-75\left(8x+9\right)$$

Multiply 216 and 1 to get 216.

$$-32x^{2}-36x+216x=-75\left(8x+9\right)$$

Combine -36x and 216x to get 180x.

$$-32x^{2}+180x=-75\left(8x+9\right)$$

Use the distributive property to multiply -75 by 8x+9.

$$-32x^{2}+180x=-600x-675$$

Add 600x to both sides.

$$-32x^{2}+180x+600x=-675$$

Combine 180x and 600x to get 780x.

$$-32x^{2}+780x=-675$$

Divide both sides by -32.

$$\frac{-32x^{2}+780x}{-32}=-\frac{675}{-32}$$

Dividing by -32 undoes the multiplication by -32.

$$x^{2}+\frac{780}{-32}x=-\frac{675}{-32}$$

Reduce the fraction \frac{780}{-32} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{195}{8}x=-\frac{675}{-32}$$

Divide -675 by -32.

$$x^{2}-\frac{195}{8}x=\frac{675}{32}$$

Divide -\frac{195}{8}, the coefficient of the x term, by 2 to get -\frac{195}{16}. Then add the square of -\frac{195}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{195}{8}x+\left(-\frac{195}{16}\right)^{2}=\frac{675}{32}+\left(-\frac{195}{16}\right)^{2}$$

Square -\frac{195}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{675}{32}+\frac{38025}{256}$$

Add \frac{675}{32} to \frac{38025}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{43425}{256}$$

Factor x^{2}-\frac{195}{8}x+\frac{38025}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{195}{16}\right)^{2}=\frac{43425}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{195}{16}\right)^{2}}=\sqrt{\frac{43425}{256}}$$

Simplify.

$$x-\frac{195}{16}=\frac{15\sqrt{193}}{16} x-\frac{195}{16}=-\frac{15\sqrt{193}}{16}$$

Add \frac{195}{16} to both sides of the equation.

$$x=\frac{15\sqrt{193}+195}{16} x=\frac{195-15\sqrt{193}}{16}$$