Solve 3/8x+1-1/x=6/7x-1

$$\frac { 3 } { 8 x + 1 } - \frac { 1 } { x } = \frac { 6 } { 7 x - 1 }$$

Solve for x

$$x=\frac{3\sqrt{11}-4}{83}\approx 0.071685233$$

$$x=\frac{-3\sqrt{11}-4}{83}\approx -0.168070776$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{1}{8},0,\frac{1}{7} since division by zero is not defined. Multiply both sides of the equation by x\left(7x-1\right)\left(8x+1\right), the least common multiple of 8x+1,x,7x-1.

$$x\left(7x-1\right)\times 3-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply x by 7x-1.

$$\left(7x^{2}-x\right)\times 3-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply 7x^{2}-x by 3.

$$21x^{2}-3x-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply 7x-1 by 8x+1 and combine like terms.

$$21x^{2}-3x-\left(56x^{2}-x-1\right)=x\left(8x+1\right)\times 6$$

To find the opposite of 56x^{2}-x-1, find the opposite of each term.

$$21x^{2}-3x-56x^{2}+x+1=x\left(8x+1\right)\times 6$$

Combine 21x^{2} and -56x^{2} to get -35x^{2}.

$$-35x^{2}-3x+x+1=x\left(8x+1\right)\times 6$$

Combine -3x and x to get -2x.

$$-35x^{2}-2x+1=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply x by 8x+1.

$$-35x^{2}-2x+1=\left(8x^{2}+x\right)\times 6$$

Use the distributive property to multiply 8x^{2}+x by 6.

$$-35x^{2}-2x+1=48x^{2}+6x$$

Subtract 48x^{2} from both sides.

$$-35x^{2}-2x+1-48x^{2}=6x$$

Combine -35x^{2} and -48x^{2} to get -83x^{2}.

$$-83x^{2}-2x+1=6x$$

Subtract 6x from both sides.

$$-83x^{2}-2x+1-6x=0$$

Combine -2x and -6x to get -8x.

$$-83x^{2}-8x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -83 for a, -8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-83\right)}}{2\left(-83\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-83\right)}}{2\left(-83\right)}$$

Multiply -4 times -83.

$$x=\frac{-\left(-8\right)±\sqrt{64+332}}{2\left(-83\right)}$$

Add 64 to 332.

$$x=\frac{-\left(-8\right)±\sqrt{396}}{2\left(-83\right)}$$

Take the square root of 396.

$$x=\frac{-\left(-8\right)±6\sqrt{11}}{2\left(-83\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±6\sqrt{11}}{2\left(-83\right)}$$

Multiply 2 times -83.

$$x=\frac{8±6\sqrt{11}}{-166}$$

Now solve the equation x=\frac{8±6\sqrt{11}}{-166} when ± is plus. Add 8 to 6\sqrt{11}.

$$x=\frac{6\sqrt{11}+8}{-166}$$

Divide 8+6\sqrt{11} by -166.

$$x=\frac{-3\sqrt{11}-4}{83}$$

Now solve the equation x=\frac{8±6\sqrt{11}}{-166} when ± is minus. Subtract 6\sqrt{11} from 8.

$$x=\frac{8-6\sqrt{11}}{-166}$$

Divide 8-6\sqrt{11} by -166.

$$x=\frac{3\sqrt{11}-4}{83}$$

The equation is now solved.

$$x=\frac{-3\sqrt{11}-4}{83} x=\frac{3\sqrt{11}-4}{83}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{1}{8},0,\frac{1}{7} since division by zero is not defined. Multiply both sides of the equation by x\left(7x-1\right)\left(8x+1\right), the least common multiple of 8x+1,x,7x-1.

$$x\left(7x-1\right)\times 3-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply x by 7x-1.

$$\left(7x^{2}-x\right)\times 3-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply 7x^{2}-x by 3.

$$21x^{2}-3x-\left(7x-1\right)\left(8x+1\right)=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply 7x-1 by 8x+1 and combine like terms.

$$21x^{2}-3x-\left(56x^{2}-x-1\right)=x\left(8x+1\right)\times 6$$

To find the opposite of 56x^{2}-x-1, find the opposite of each term.

$$21x^{2}-3x-56x^{2}+x+1=x\left(8x+1\right)\times 6$$

Combine 21x^{2} and -56x^{2} to get -35x^{2}.

$$-35x^{2}-3x+x+1=x\left(8x+1\right)\times 6$$

Combine -3x and x to get -2x.

$$-35x^{2}-2x+1=x\left(8x+1\right)\times 6$$

Use the distributive property to multiply x by 8x+1.

$$-35x^{2}-2x+1=\left(8x^{2}+x\right)\times 6$$

Use the distributive property to multiply 8x^{2}+x by 6.

$$-35x^{2}-2x+1=48x^{2}+6x$$

Subtract 48x^{2} from both sides.

$$-35x^{2}-2x+1-48x^{2}=6x$$

Combine -35x^{2} and -48x^{2} to get -83x^{2}.

$$-83x^{2}-2x+1=6x$$

Subtract 6x from both sides.

$$-83x^{2}-2x+1-6x=0$$

Combine -2x and -6x to get -8x.

$$-83x^{2}-8x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-83x^{2}-8x=-1$$

Divide both sides by -83.

$$\frac{-83x^{2}-8x}{-83}=-\frac{1}{-83}$$

Dividing by -83 undoes the multiplication by -83.

$$x^{2}+\left(-\frac{8}{-83}\right)x=-\frac{1}{-83}$$

Divide -8 by -83.

$$x^{2}+\frac{8}{83}x=-\frac{1}{-83}$$

Divide -1 by -83.

$$x^{2}+\frac{8}{83}x=\frac{1}{83}$$

Divide \frac{8}{83}, the coefficient of the x term, by 2 to get \frac{4}{83}. Then add the square of \frac{4}{83} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{83}x+\left(\frac{4}{83}\right)^{2}=\frac{1}{83}+\left(\frac{4}{83}\right)^{2}$$

Square \frac{4}{83} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{83}x+\frac{16}{6889}=\frac{1}{83}+\frac{16}{6889}$$

Add \frac{1}{83} to \frac{16}{6889} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{83}x+\frac{16}{6889}=\frac{99}{6889}$$

Factor x^{2}+\frac{8}{83}x+\frac{16}{6889}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4}{83}\right)^{2}=\frac{99}{6889}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{83}\right)^{2}}=\sqrt{\frac{99}{6889}}$$

Simplify.

$$x+\frac{4}{83}=\frac{3\sqrt{11}}{83} x+\frac{4}{83}=-\frac{3\sqrt{11}}{83}$$

Subtract \frac{4}{83} from both sides of the equation.

$$x=\frac{3\sqrt{11}-4}{83} x=\frac{-3\sqrt{11}-4}{83}$$