Solve 2240/280-x=8+10/9x%

$$\frac{ 2240 }{ 280-x } =8+ \frac{ 10 }{ 9 } x \%$$

Solve for x

$$x=-440$$

$$x=0$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 280 since division by zero is not defined. Multiply both sides of the equation by 900\left(x-280\right), the least common multiple of 280-x,9,100.

$$-900\times 2240=900\left(x-280\right)\times 8+\frac{10}{9}\left(9x-2520\right)x$$

Multiply -900 and 2240 to get -2016000.

$$-2016000=900\left(x-280\right)\times 8+\frac{10}{9}\left(9x-2520\right)x$$

Multiply 900 and 8 to get 7200.

$$-2016000=7200\left(x-280\right)+\frac{10}{9}\left(9x-2520\right)x$$

Use the distributive property to multiply 7200 by x-280.

$$-2016000=7200x-2016000+\frac{10}{9}\left(9x-2520\right)x$$

Use the distributive property to multiply \frac{10}{9} by 9x-2520.

$$-2016000=7200x-2016000+\left(10x-2800\right)x$$

Use the distributive property to multiply 10x-2800 by x.

$$-2016000=7200x-2016000+10x^{2}-2800x$$

Combine 7200x and -2800x to get 4400x.

$$-2016000=4400x-2016000+10x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4400x-2016000+10x^{2}=-2016000$$

Add 2016000 to both sides.

$$4400x-2016000+10x^{2}+2016000=0$$

Add -2016000 and 2016000 to get 0.

$$4400x+10x^{2}=0$$

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.

$$10x^{2}+4400x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 4400 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4400±\sqrt{4400^{2}}}{2\times 10}$$

Take the square root of 4400^{2}.

$$x=\frac{-4400±4400}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-4400±4400}{20}$$

Now solve the equation x=\frac{-4400±4400}{20} when ± is plus. Add -4400 to 4400.

$$x=\frac{0}{20}$$

Divide 0 by 20.

$$x=0$$

Now solve the equation x=\frac{-4400±4400}{20} when ± is minus. Subtract 4400 from -4400.

$$x=-\frac{8800}{20}$$

Divide -8800 by 20.

$$x=-440$$

The equation is now solved.

$$x=0 x=-440$$

Steps for Completing the Square

Variable x cannot be equal to 280 since division by zero is not defined. Multiply both sides of the equation by 900\left(x-280\right), the least common multiple of 280-x,9,100.

$$-900\times 2240=900\left(x-280\right)\times 8+\frac{10}{9}\left(9x-2520\right)x$$

Multiply -900 and 2240 to get -2016000.

$$-2016000=900\left(x-280\right)\times 8+\frac{10}{9}\left(9x-2520\right)x$$

Multiply 900 and 8 to get 7200.

$$-2016000=7200\left(x-280\right)+\frac{10}{9}\left(9x-2520\right)x$$

Use the distributive property to multiply 7200 by x-280.

$$-2016000=7200x-2016000+\frac{10}{9}\left(9x-2520\right)x$$

Use the distributive property to multiply \frac{10}{9} by 9x-2520.

$$-2016000=7200x-2016000+\left(10x-2800\right)x$$

Use the distributive property to multiply 10x-2800 by x.

$$-2016000=7200x-2016000+10x^{2}-2800x$$

Combine 7200x and -2800x to get 4400x.

$$-2016000=4400x-2016000+10x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4400x-2016000+10x^{2}=-2016000$$

Add 2016000 to both sides.

$$4400x+10x^{2}=-2016000+2016000$$

Add -2016000 and 2016000 to get 0.

$$4400x+10x^{2}=0$$

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.

$$10x^{2}+4400x=0$$

Divide both sides by 10.

$$\frac{10x^{2}+4400x}{10}=\frac{0}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{4400}{10}x=\frac{0}{10}$$

Divide 4400 by 10.

$$x^{2}+440x=\frac{0}{10}$$

Divide 0 by 10.

$$x^{2}+440x=0$$

Divide 440, the coefficient of the x term, by 2 to get 220. Then add the square of 220 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+440x+220^{2}=220^{2}$$

Square 220.

$$x^{2}+440x+48400=48400$$

Factor x^{2}+440x+48400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+220\right)^{2}=48400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+220\right)^{2}}=\sqrt{48400}$$

Simplify.

$$x+220=220 x+220=-220$$

Subtract 220 from both sides of the equation.

$$x=0 x=-440$$