Solve x-1/x(8x/1)+1/8(8x)/1=frac{-5}{x}(8x)

$$\frac { x - 1 } { x } ( \frac { 8 x } { 1 } ) + \frac { 1 } { 8 } \frac { ( 8 x ) } { 1 } = \frac { - 5 } { x } ( 8 x )$$

Solve for x

$$x = -\frac{32}{9} = -3\frac{5}{9} \approx -3.555555556$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8x, the least common multiple of x,8.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times 8x\times 8x=8\left(-5\right)\times 8x$$

Multiply 8x and 8x to get \left(8x\right)^{2}.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Anything divided by one gives itself.

$$8\left(x-1\right)\times 8x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Multiply 8 and 8 to get 64.

$$64\left(x-1\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64 by x-1.

$$\left(64x-64\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64x-64 by x.

$$64x^{2}-64x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Expand \left(8x\right)^{2}.

$$64x^{2}-64x+\frac{1}{8}\times 8^{2}x^{2}=8\left(-5\right)\times 8x$$

Calculate 8 to the power of 2 and get 64.

$$64x^{2}-64x+\frac{1}{8}\times 64x^{2}=8\left(-5\right)\times 8x$$

Multiply \frac{1}{8} and 64 to get \frac{64}{8}.

$$64x^{2}-64x+\frac{64}{8}x^{2}=8\left(-5\right)\times 8x$$

Divide 64 by 8 to get 8.

$$64x^{2}-64x+8x^{2}=8\left(-5\right)\times 8x$$

Combine 64x^{2} and 8x^{2} to get 72x^{2}.

$$72x^{2}-64x=8\left(-5\right)\times 8x$$

Multiply 8 and -5 to get -40.

$$72x^{2}-64x=-40\times 8x$$

Multiply -40 and 8 to get -320.

$$72x^{2}-64x=-320x$$

Add 320x to both sides.

$$72x^{2}-64x+320x=0$$

Combine -64x and 320x to get 256x.

$$72x^{2}+256x=0$$

Factor out x.

$$x\left(72x+256\right)=0$$

To find equation solutions, solve x=0 and 72x+256=0.

$$x=0 x=-\frac{32}{9}$$

Variable x cannot be equal to 0.

$$x=-\frac{32}{9}$$

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 8x, the least common multiple of x,8.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times 8x\times 8x=8\left(-5\right)\times 8x$$

Multiply 8x and 8x to get \left(8x\right)^{2}.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Anything divided by one gives itself.

$$8\left(x-1\right)\times 8x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Multiply 8 and 8 to get 64.

$$64\left(x-1\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64 by x-1.

$$\left(64x-64\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64x-64 by x.

$$64x^{2}-64x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Expand \left(8x\right)^{2}.

$$64x^{2}-64x+\frac{1}{8}\times 8^{2}x^{2}=8\left(-5\right)\times 8x$$

Calculate 8 to the power of 2 and get 64.

$$64x^{2}-64x+\frac{1}{8}\times 64x^{2}=8\left(-5\right)\times 8x$$

Multiply \frac{1}{8} and 64 to get \frac{64}{8}.

$$64x^{2}-64x+\frac{64}{8}x^{2}=8\left(-5\right)\times 8x$$

Divide 64 by 8 to get 8.

$$64x^{2}-64x+8x^{2}=8\left(-5\right)\times 8x$$

Combine 64x^{2} and 8x^{2} to get 72x^{2}.

$$72x^{2}-64x=8\left(-5\right)\times 8x$$

Multiply 8 and -5 to get -40.

$$72x^{2}-64x=-40\times 8x$$

Multiply -40 and 8 to get -320.

$$72x^{2}-64x=-320x$$

Add 320x to both sides.

$$72x^{2}-64x+320x=0$$

Combine -64x and 320x to get 256x.

$$72x^{2}+256x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 72 for a, 256 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-256±\sqrt{256^{2}}}{2\times 72}$$

Take the square root of 256^{2}.

$$x=\frac{-256±256}{2\times 72}$$

Multiply 2 times 72.

$$x=\frac{-256±256}{144}$$

Now solve the equation x=\frac{-256±256}{144} when ± is plus. Add -256 to 256.

$$x=\frac{0}{144}$$

Divide 0 by 144.

$$x=0$$

Now solve the equation x=\frac{-256±256}{144} when ± is minus. Subtract 256 from -256.

$$x=-\frac{512}{144}$$

Reduce the fraction \frac{-512}{144} to lowest terms by extracting and canceling out 16.

$$x=-\frac{32}{9}$$

The equation is now solved.

$$x=0 x=-\frac{32}{9}$$

Variable x cannot be equal to 0.

$$x=-\frac{32}{9}$$

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 8x, the least common multiple of x,8.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times 8x\times 8x=8\left(-5\right)\times 8x$$

Multiply 8x and 8x to get \left(8x\right)^{2}.

$$8\left(x-1\right)\times \frac{8x}{1}+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Anything divided by one gives itself.

$$8\left(x-1\right)\times 8x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Multiply 8 and 8 to get 64.

$$64\left(x-1\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64 by x-1.

$$\left(64x-64\right)x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Use the distributive property to multiply 64x-64 by x.

$$64x^{2}-64x+\frac{1}{8}\times \left(8x\right)^{2}=8\left(-5\right)\times 8x$$

Expand \left(8x\right)^{2}.

$$64x^{2}-64x+\frac{1}{8}\times 8^{2}x^{2}=8\left(-5\right)\times 8x$$

Calculate 8 to the power of 2 and get 64.

$$64x^{2}-64x+\frac{1}{8}\times 64x^{2}=8\left(-5\right)\times 8x$$

Multiply \frac{1}{8} and 64 to get \frac{64}{8}.

$$64x^{2}-64x+\frac{64}{8}x^{2}=8\left(-5\right)\times 8x$$

Divide 64 by 8 to get 8.

$$64x^{2}-64x+8x^{2}=8\left(-5\right)\times 8x$$

Combine 64x^{2} and 8x^{2} to get 72x^{2}.

$$72x^{2}-64x=8\left(-5\right)\times 8x$$

Multiply 8 and -5 to get -40.

$$72x^{2}-64x=-40\times 8x$$

Multiply -40 and 8 to get -320.

$$72x^{2}-64x=-320x$$

Add 320x to both sides.

$$72x^{2}-64x+320x=0$$

Combine -64x and 320x to get 256x.

$$72x^{2}+256x=0$$

Divide both sides by 72.

$$\frac{72x^{2}+256x}{72}=\frac{0}{72}$$

Dividing by 72 undoes the multiplication by 72.

$$x^{2}+\frac{256}{72}x=\frac{0}{72}$$

Reduce the fraction \frac{256}{72} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{32}{9}x=\frac{0}{72}$$

Divide 0 by 72.

$$x^{2}+\frac{32}{9}x=0$$

Divide \frac{32}{9}, the coefficient of the x term, by 2 to get \frac{16}{9}. Then add the square of \frac{16}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{32}{9}x+\left(\frac{16}{9}\right)^{2}=\left(\frac{16}{9}\right)^{2}$$

Square \frac{16}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{32}{9}x+\frac{256}{81}=\frac{256}{81}$$

Factor x^{2}+\frac{32}{9}x+\frac{256}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{16}{9}\right)^{2}=\frac{256}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{16}{9}\right)^{2}}=\sqrt{\frac{256}{81}}$$

Simplify.

$$x+\frac{16}{9}=\frac{16}{9} x+\frac{16}{9}=-\frac{16}{9}$$

Subtract \frac{16}{9} from both sides of the equation.

$$x=0 x=-\frac{32}{9}$$

Variable x cannot be equal to 0.

$$x=-\frac{32}{9}$$