Solve 255/x+1+255/x=32

$$\frac{ 255 }{ x+1 } + \frac{ 255 }{ x } =32$$

Solve for x

$$x = \frac{\sqrt{65281} + 239}{32} \approx 15.453170866$$

$$x=\frac{239-\sqrt{65281}}{32}\approx -0.515670866$$

Steps Using the Quadratic Formula

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

$$x\times 255+\left(x+1\right)\times 255=32x\left(x+1\right)$$

Use the distributive property to multiply x+1 by 255.

$$x\times 255+255x+255=32x\left(x+1\right)$$

Combine x\times 255 and 255x to get 510x.

$$510x+255=32x\left(x+1\right)$$

Use the distributive property to multiply 32x by x+1.

$$510x+255=32x^{2}+32x$$

Subtract 32x^{2} from both sides.

$$510x+255-32x^{2}=32x$$

Subtract 32x from both sides.

$$510x+255-32x^{2}-32x=0$$

Combine 510x and -32x to get 478x.

$$478x+255-32x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-32x^{2}+478x+255=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, 478 for b, and 255 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-478±\sqrt{478^{2}-4\left(-32\right)\times 255}}{2\left(-32\right)}$$

Square 478.

$$x=\frac{-478±\sqrt{228484-4\left(-32\right)\times 255}}{2\left(-32\right)}$$

Multiply -4 times -32.

$$x=\frac{-478±\sqrt{228484+128\times 255}}{2\left(-32\right)}$$

Multiply 128 times 255.

$$x=\frac{-478±\sqrt{228484+32640}}{2\left(-32\right)}$$

Add 228484 to 32640.

$$x=\frac{-478±\sqrt{261124}}{2\left(-32\right)}$$

Take the square root of 261124.

$$x=\frac{-478±2\sqrt{65281}}{2\left(-32\right)}$$

Multiply 2 times -32.

$$x=\frac{-478±2\sqrt{65281}}{-64}$$

Now solve the equation x=\frac{-478±2\sqrt{65281}}{-64} when ± is plus. Add -478 to 2\sqrt{65281}.

$$x=\frac{2\sqrt{65281}-478}{-64}$$

Divide -478+2\sqrt{65281} by -64.

$$x=\frac{239-\sqrt{65281}}{32}$$

Now solve the equation x=\frac{-478±2\sqrt{65281}}{-64} when ± is minus. Subtract 2\sqrt{65281} from -478.

$$x=\frac{-2\sqrt{65281}-478}{-64}$$

Divide -478-2\sqrt{65281} by -64.

$$x=\frac{\sqrt{65281}+239}{32}$$

The equation is now solved.

$$x=\frac{239-\sqrt{65281}}{32} x=\frac{\sqrt{65281}+239}{32}$$

Steps for Completing the Square

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

$$x\times 255+\left(x+1\right)\times 255=32x\left(x+1\right)$$

Use the distributive property to multiply x+1 by 255.

$$x\times 255+255x+255=32x\left(x+1\right)$$

Combine x\times 255 and 255x to get 510x.

$$510x+255=32x\left(x+1\right)$$

Use the distributive property to multiply 32x by x+1.

$$510x+255=32x^{2}+32x$$

Subtract 32x^{2} from both sides.

$$510x+255-32x^{2}=32x$$

Subtract 32x from both sides.

$$510x+255-32x^{2}-32x=0$$

Combine 510x and -32x to get 478x.

$$478x+255-32x^{2}=0$$

Subtract 255 from both sides. Anything subtracted from zero gives its negation.

$$478x-32x^{2}=-255$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-32x^{2}+478x=-255$$

Divide both sides by -32.

$$\frac{-32x^{2}+478x}{-32}=-\frac{255}{-32}$$

Dividing by -32 undoes the multiplication by -32.

$$x^{2}+\frac{478}{-32}x=-\frac{255}{-32}$$

Reduce the fraction \frac{478}{-32} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{239}{16}x=-\frac{255}{-32}$$

Divide -255 by -32.

$$x^{2}-\frac{239}{16}x=\frac{255}{32}$$

Divide -\frac{239}{16}, the coefficient of the x term, by 2 to get -\frac{239}{32}. Then add the square of -\frac{239}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{239}{16}x+\left(-\frac{239}{32}\right)^{2}=\frac{255}{32}+\left(-\frac{239}{32}\right)^{2}$$

Square -\frac{239}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{239}{16}x+\frac{57121}{1024}=\frac{255}{32}+\frac{57121}{1024}$$

Add \frac{255}{32} to \frac{57121}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{239}{16}x+\frac{57121}{1024}=\frac{65281}{1024}$$

Factor x^{2}-\frac{239}{16}x+\frac{57121}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{239}{32}\right)^{2}=\frac{65281}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{239}{32}\right)^{2}}=\sqrt{\frac{65281}{1024}}$$

Simplify.

$$x-\frac{239}{32}=\frac{\sqrt{65281}}{32} x-\frac{239}{32}=-\frac{\sqrt{65281}}{32}$$

Add \frac{239}{32} to both sides of the equation.

$$x=\frac{\sqrt{65281}+239}{32} x=\frac{239-\sqrt{65281}}{32}$$