Solve 2800/x+10=2730/x+1

$$\frac{ 2800 }{ x } +10= \frac{ 2730 }{ x+1 }$$

Solve for x (complex solution)

$$x=-4+2\sqrt{66}i\approx -4+16.248076809i$$

$$x=-2\sqrt{66}i-4\approx -4-16.248076809i$$

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,x+1.

$$\left(x+1\right)\times 2800+x\left(x+1\right)\times 10=x\times 2730$$

Use the distributive property to multiply x+1 by 2800.

$$2800x+2800+x\left(x+1\right)\times 10=x\times 2730$$

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

$$2800x+2800+\left(x^{2}+x\right)\times 10=x\times 2730$$

Use the distributive property to multiply x^{2}+x by 10.

$$2800x+2800+10x^{2}+10x=x\times 2730$$

Combine 2800x and 10x to get 2810x.

$$2810x+2800+10x^{2}=x\times 2730$$

Subtract x\times 2730 from both sides.

$$2810x+2800+10x^{2}-x\times 2730=0$$

Combine 2810x and -x\times 2730 to get 80x.

$$80x+2800+10x^{2}=0$$

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.

$$10x^{2}+80x+2800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 80 for b, and 2800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times 10\times 2800}}{2\times 10}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\times 10\times 2800}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-80±\sqrt{6400-40\times 2800}}{2\times 10}$$

Multiply -40 times 2800.

$$x=\frac{-80±\sqrt{6400-112000}}{2\times 10}$$

Add 6400 to -112000.

$$x=\frac{-80±\sqrt{-105600}}{2\times 10}$$

Take the square root of -105600.

$$x=\frac{-80±40\sqrt{66}i}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-80±40\sqrt{66}i}{20}$$

Now solve the equation x=\frac{-80±40\sqrt{66}i}{20} when ± is plus. Add -80 to 40i\sqrt{66}.

$$x=\frac{-80+40\sqrt{66}i}{20}$$

Divide -80+40i\sqrt{66} by 20.

$$x=-4+2\sqrt{66}i$$

Now solve the equation x=\frac{-80±40\sqrt{66}i}{20} when ± is minus. Subtract 40i\sqrt{66} from -80.

$$x=\frac{-40\sqrt{66}i-80}{20}$$

Divide -80-40i\sqrt{66} by 20.

$$x=-2\sqrt{66}i-4$$

The equation is now solved.

$$x=-4+2\sqrt{66}i x=-2\sqrt{66}i-4$$

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,x+1.

$$\left(x+1\right)\times 2800+x\left(x+1\right)\times 10=x\times 2730$$

Use the distributive property to multiply x+1 by 2800.

$$2800x+2800+x\left(x+1\right)\times 10=x\times 2730$$

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

$$2800x+2800+\left(x^{2}+x\right)\times 10=x\times 2730$$

Use the distributive property to multiply x^{2}+x by 10.

$$2800x+2800+10x^{2}+10x=x\times 2730$$

Combine 2800x and 10x to get 2810x.

$$2810x+2800+10x^{2}=x\times 2730$$

Subtract x\times 2730 from both sides.

$$2810x+2800+10x^{2}-x\times 2730=0$$

Combine 2810x and -x\times 2730 to get 80x.

$$80x+2800+10x^{2}=0$$

Subtract 2800 from both sides. Anything subtracted from zero gives its negation.

$$80x+10x^{2}=-2800$$

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.

$$10x^{2}+80x=-2800$$

Divide both sides by 10.

$$\frac{10x^{2}+80x}{10}=-\frac{2800}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{80}{10}x=-\frac{2800}{10}$$

Divide 80 by 10.

$$x^{2}+8x=-\frac{2800}{10}$$

Divide -2800 by 10.

$$x^{2}+8x=-280$$

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

$$x^{2}+8x+4^{2}=-280+4^{2}$$

Square 4.

$$x^{2}+8x+16=-280+16$$

Add -280 to 16.

$$x^{2}+8x+16=-264$$

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

$$\left(x+4\right)^{2}=-264$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{-264}$$

Simplify.

$$x+4=2\sqrt{66}i x+4=-2\sqrt{66}i$$

Subtract 4 from both sides of the equation.

$$x=-4+2\sqrt{66}i x=-2\sqrt{66}i-4$$