Solve 450/x+0.5=440/x+10

$$\frac{ 450 }{ x } +0.5 = \frac{ 440 }{ x+10 }$$

Solve for x (complex solution)

$$x=-15+i\times 15\sqrt{39}\approx -15+93.674969976i$$

$$x=-i\times 15\sqrt{39}-15\approx -15-93.674969976i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x,x+10.

$$\left(x+10\right)\times 450+x\left(x+10\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x+10 by 450.

$$450x+4500+x\left(x+10\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x by x+10.

$$450x+4500+\left(x^{2}+10x\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x^{2}+10x by 0.5.

$$450x+4500+0.5x^{2}+5x=x\times 440$$

Combine 450x and 5x to get 455x.

$$455x+4500+0.5x^{2}=x\times 440$$

Subtract x\times 440 from both sides.

$$455x+4500+0.5x^{2}-x\times 440=0$$

Combine 455x and -x\times 440 to get 15x.

$$15x+4500+0.5x^{2}=0$$

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.

$$0.5x^{2}+15x+4500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.5 for a, 15 for b, and 4500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 0.5\times 4500}}{2\times 0.5}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 0.5\times 4500}}{2\times 0.5}$$

Multiply -4 times 0.5.

$$x=\frac{-15±\sqrt{225-2\times 4500}}{2\times 0.5}$$

Multiply -2 times 4500.

$$x=\frac{-15±\sqrt{225-9000}}{2\times 0.5}$$

Add 225 to -9000.

$$x=\frac{-15±\sqrt{-8775}}{2\times 0.5}$$

Take the square root of -8775.

$$x=\frac{-15±15\sqrt{39}i}{2\times 0.5}$$

Multiply 2 times 0.5.

$$x=\frac{-15±15\sqrt{39}i}{1}$$

Now solve the equation x=\frac{-15±15\sqrt{39}i}{1} when ± is plus. Add -15 to 15i\sqrt{39}.

$$x=\frac{-15+15\sqrt{39}i}{1}$$

Divide -15+15i\sqrt{39} by 1.

$$x=-15+15\sqrt{39}i$$

Now solve the equation x=\frac{-15±15\sqrt{39}i}{1} when ± is minus. Subtract 15i\sqrt{39} from -15.

$$x=\frac{-15\sqrt{39}i-15}{1}$$

Divide -15-15i\sqrt{39} by 1.

$$x=-15\sqrt{39}i-15$$

The equation is now solved.

$$x=-15+15\sqrt{39}i x=-15\sqrt{39}i-15$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+10\right), the least common multiple of x,x+10.

$$\left(x+10\right)\times 450+x\left(x+10\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x+10 by 450.

$$450x+4500+x\left(x+10\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x by x+10.

$$450x+4500+\left(x^{2}+10x\right)\times 0.5=x\times 440$$

Use the distributive property to multiply x^{2}+10x by 0.5.

$$450x+4500+0.5x^{2}+5x=x\times 440$$

Combine 450x and 5x to get 455x.

$$455x+4500+0.5x^{2}=x\times 440$$

Subtract x\times 440 from both sides.

$$455x+4500+0.5x^{2}-x\times 440=0$$

Combine 455x and -x\times 440 to get 15x.

$$15x+4500+0.5x^{2}=0$$

Subtract 4500 from both sides. Anything subtracted from zero gives its negation.

$$15x+0.5x^{2}=-4500$$

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.

$$0.5x^{2}+15x=-4500$$

Multiply both sides by 2.

$$\frac{0.5x^{2}+15x}{0.5}=-\frac{4500}{0.5}$$

Dividing by 0.5 undoes the multiplication by 0.5.

$$x^{2}+\frac{15}{0.5}x=-\frac{4500}{0.5}$$

Divide 15 by 0.5 by multiplying 15 by the reciprocal of 0.5.

$$x^{2}+30x=-\frac{4500}{0.5}$$

Divide -4500 by 0.5 by multiplying -4500 by the reciprocal of 0.5.

$$x^{2}+30x=-9000$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=-9000+15^{2}$$

Square 15.

$$x^{2}+30x+225=-9000+225$$

Add -9000 to 225.

$$x^{2}+30x+225=-8775$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=-8775$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{-8775}$$

Simplify.

$$x+15=15\sqrt{39}i x+15=-15\sqrt{39}i$$

Subtract 15 from both sides of the equation.

$$x=-15+15\sqrt{39}i x=-15\sqrt{39}i-15$$