Solve 100000/x+1+1000=120000/x

$$\frac { 100000 } { x + 1 } + 1000 = \frac { 120000 } { x }$$

Solve for x

$$x=-5$$

$$x=24$$

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.

$$x\times 100000+x\left(x+1\right)\times 1000=\left(x+1\right)\times 120000$$

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

$$x\times 100000+\left(x^{2}+x\right)\times 1000=\left(x+1\right)\times 120000$$

Use the distributive property to multiply x^{2}+x by 1000.

$$x\times 100000+1000x^{2}+1000x=\left(x+1\right)\times 120000$$

Combine x\times 100000 and 1000x to get 101000x.

$$101000x+1000x^{2}=\left(x+1\right)\times 120000$$

Use the distributive property to multiply x+1 by 120000.

$$101000x+1000x^{2}=120000x+120000$$

Subtract 120000x from both sides.

$$101000x+1000x^{2}-120000x=120000$$

Combine 101000x and -120000x to get -19000x.

$$-19000x+1000x^{2}=120000$$

Subtract 120000 from both sides.

$$-19000x+1000x^{2}-120000=0$$

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.

$$1000x^{2}-19000x-120000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, -19000 for b, and -120000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19000\right)±\sqrt{\left(-19000\right)^{2}-4\times 1000\left(-120000\right)}}{2\times 1000}$$

Square -19000.

$$x=\frac{-\left(-19000\right)±\sqrt{361000000-4\times 1000\left(-120000\right)}}{2\times 1000}$$

Multiply -4 times 1000.

$$x=\frac{-\left(-19000\right)±\sqrt{361000000-4000\left(-120000\right)}}{2\times 1000}$$

Multiply -4000 times -120000.

$$x=\frac{-\left(-19000\right)±\sqrt{361000000+480000000}}{2\times 1000}$$

Add 361000000 to 480000000.

$$x=\frac{-\left(-19000\right)±\sqrt{841000000}}{2\times 1000}$$

Take the square root of 841000000.

$$x=\frac{-\left(-19000\right)±29000}{2\times 1000}$$

The opposite of -19000 is 19000.

$$x=\frac{19000±29000}{2\times 1000}$$

Multiply 2 times 1000.

$$x=\frac{19000±29000}{2000}$$

Now solve the equation x=\frac{19000±29000}{2000} when ± is plus. Add 19000 to 29000.

$$x=\frac{48000}{2000}$$

Divide 48000 by 2000.

$$x=24$$

Now solve the equation x=\frac{19000±29000}{2000} when ± is minus. Subtract 29000 from 19000.

$$x=-\frac{10000}{2000}$$

Divide -10000 by 2000.

$$x=-5$$

The equation is now solved.

$$x=24 x=-5$$

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.

$$x\times 100000+x\left(x+1\right)\times 1000=\left(x+1\right)\times 120000$$

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

$$x\times 100000+\left(x^{2}+x\right)\times 1000=\left(x+1\right)\times 120000$$

Use the distributive property to multiply x^{2}+x by 1000.

$$x\times 100000+1000x^{2}+1000x=\left(x+1\right)\times 120000$$

Combine x\times 100000 and 1000x to get 101000x.

$$101000x+1000x^{2}=\left(x+1\right)\times 120000$$

Use the distributive property to multiply x+1 by 120000.

$$101000x+1000x^{2}=120000x+120000$$

Subtract 120000x from both sides.

$$101000x+1000x^{2}-120000x=120000$$

Combine 101000x and -120000x to get -19000x.

$$-19000x+1000x^{2}=120000$$

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.

$$1000x^{2}-19000x=120000$$

Divide both sides by 1000.

$$\frac{1000x^{2}-19000x}{1000}=\frac{120000}{1000}$$

Dividing by 1000 undoes the multiplication by 1000.

$$x^{2}+\left(-\frac{19000}{1000}\right)x=\frac{120000}{1000}$$

Divide -19000 by 1000.

$$x^{2}-19x=\frac{120000}{1000}$$

Divide 120000 by 1000.

$$x^{2}-19x=120$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=120+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=120+\frac{361}{4}$$

Add 120 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{841}{4}$$

Factor x^{2}-19x+\frac{361}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{29}{2} x-\frac{19}{2}=-\frac{29}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=24 x=-5$$