Solve (10+(1+5/2x%))*(50*(1-1/3x%)-30)=250

$$(10+(1+ \frac{ 5 }{ 2 } x \% )) \times (50 \times (1- \frac{ 1 }{ 3 } x \% )-30)=250$$

Solve for x

$$x=20\sqrt{46}-160\approx -24.353400337$$

$$x=-20\sqrt{46}-160\approx -295.646599663$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 300, the least common multiple of 2,100,3.

$$300\left(10+1+\frac{5}{2}\times \frac{x}{100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Add 10 and 1 to get 11.

$$300\left(11+\frac{5}{2}\times \frac{x}{100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Multiply \frac{5}{2} times \frac{x}{100} by multiplying numerator times numerator and denominator times denominator.

$$300\left(11+\frac{5x}{2\times 100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Cancel out 5 in both numerator and denominator.

$$300\left(11+\frac{x}{2\times 20}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11 times \frac{2\times 20}{2\times 20}.

$$300\left(\frac{11\times 2\times 20}{2\times 20}+\frac{x}{2\times 20}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Since \frac{11\times 2\times 20}{2\times 20} and \frac{x}{2\times 20} have the same denominator, add them by adding their numerators.

$$300\times \frac{11\times 2\times 20+x}{2\times 20}\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Do the multiplications in 11\times 2\times 20+x.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Multiply \frac{1}{3} times \frac{x}{100} by multiplying numerator times numerator and denominator times denominator.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{x}{3\times 100}\right)-30\right)=75000$$

Multiply 3 and 100 to get 300.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{x}{300}\right)-30\right)=75000$$

Use the distributive property to multiply 50 by 1-\frac{x}{300}.

$$300\times \frac{440+x}{2\times 20}\left(50+50\left(-\frac{x}{300}\right)-30\right)=75000$$

Cancel out 300, the greatest common factor in 50 and 300.

$$300\times \frac{440+x}{2\times 20}\left(50+\frac{x}{-6}-30\right)=75000$$

Subtract 30 from 50 to get 20.

$$300\times \frac{440+x}{2\times 20}\left(20+\frac{x}{-6}\right)=75000$$

Express 300\times \frac{440+x}{2\times 20} as a single fraction.

$$\frac{300\left(440+x\right)}{2\times 20}\left(20+\frac{x}{-6}\right)=75000$$

Cancel out 2\times 10 in both numerator and denominator.

$$\frac{15\left(x+440\right)}{2}\left(20+\frac{x}{-6}\right)=75000$$

Use the distributive property to multiply \frac{15\left(x+440\right)}{2} by 20+\frac{x}{-6}.

$$20\times \frac{15\left(x+440\right)}{2}+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 15 by x+440.

$$20\times \frac{15x+6600}{2}+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Cancel out 2, the greatest common factor in 20 and 2.

$$10\left(15x+6600\right)+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 10 by 15x+6600.

$$150x+66000+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 15 by x+440.

$$150x+66000+\frac{15x+6600}{2}\times \frac{x}{-6}=75000$$

Multiply \frac{15x+6600}{2} times \frac{x}{-6} by multiplying numerator times numerator and denominator times denominator.

$$150x+66000+\frac{\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 150x+66000 times \frac{2\left(-6\right)}{2\left(-6\right)}.

$$\frac{\left(150x+66000\right)\times 2\left(-6\right)}{2\left(-6\right)}+\frac{\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

Since \frac{\left(150x+66000\right)\times 2\left(-6\right)}{2\left(-6\right)} and \frac{\left(15x+6600\right)x}{2\left(-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(150x+66000\right)\times 2\left(-6\right)+\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

Do the multiplications in \left(150x+66000\right)\times 2\left(-6\right)+\left(15x+6600\right)x.

$$\frac{-1800x-792000+15x^{2}+6600x}{2\left(-6\right)}=75000$$

Combine like terms in -1800x-792000+15x^{2}+6600x.

$$\frac{4800x-792000+15x^{2}}{2\left(-6\right)}=75000$$

Factor the expressions that are not already factored in \frac{4800x-792000+15x^{2}}{2\left(-6\right)}.

$$\frac{15\left(x-120\right)\left(x+440\right)}{-6\times 2}=75000$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\left(x-120\right)\left(x+440\right)}{-2\times 2}=75000$$

Multiply -2 and 2 to get -4.

$$\frac{5\left(x-120\right)\left(x+440\right)}{-4}=75000$$

Use the distributive property to multiply 5 by x-120.

$$\frac{\left(5x-600\right)\left(x+440\right)}{-4}=75000$$

Apply the distributive property by multiplying each term of 5x-600 by each term of x+440.

$$\frac{5x^{2}+2200x-600x-264000}{-4}=75000$$

Combine 2200x and -600x to get 1600x.

$$\frac{5x^{2}+1600x-264000}{-4}=75000$$

Divide each term of 5x^{2}+1600x-264000 by -4 to get -\frac{5}{4}x^{2}-400x+66000.

$$-\frac{5}{4}x^{2}-400x+66000=75000$$

Subtract 75000 from both sides.

$$-\frac{5}{4}x^{2}-400x+66000-75000=0$$

Subtract 75000 from 66000 to get -9000.

$$-\frac{5}{4}x^{2}-400x-9000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{5}{4} for a, -400 for b, and -9000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\left(-\frac{5}{4}\right)\left(-9000\right)}}{2\left(-\frac{5}{4}\right)}$$

Square -400.

$$x=\frac{-\left(-400\right)±\sqrt{160000-4\left(-\frac{5}{4}\right)\left(-9000\right)}}{2\left(-\frac{5}{4}\right)}$$

Multiply -4 times -\frac{5}{4}.

$$x=\frac{-\left(-400\right)±\sqrt{160000+5\left(-9000\right)}}{2\left(-\frac{5}{4}\right)}$$

Multiply 5 times -9000.

$$x=\frac{-\left(-400\right)±\sqrt{160000-45000}}{2\left(-\frac{5}{4}\right)}$$

Add 160000 to -45000.

$$x=\frac{-\left(-400\right)±\sqrt{115000}}{2\left(-\frac{5}{4}\right)}$$

Take the square root of 115000.

$$x=\frac{-\left(-400\right)±50\sqrt{46}}{2\left(-\frac{5}{4}\right)}$$

The opposite of -400 is 400.

$$x=\frac{400±50\sqrt{46}}{2\left(-\frac{5}{4}\right)}$$

Multiply 2 times -\frac{5}{4}.

$$x=\frac{400±50\sqrt{46}}{-\frac{5}{2}}$$

Now solve the equation x=\frac{400±50\sqrt{46}}{-\frac{5}{2}} when ± is plus. Add 400 to 50\sqrt{46}.

$$x=\frac{50\sqrt{46}+400}{-\frac{5}{2}}$$

Divide 400+50\sqrt{46} by -\frac{5}{2} by multiplying 400+50\sqrt{46} by the reciprocal of -\frac{5}{2}.

$$x=-20\sqrt{46}-160$$

Now solve the equation x=\frac{400±50\sqrt{46}}{-\frac{5}{2}} when ± is minus. Subtract 50\sqrt{46} from 400.

$$x=\frac{400-50\sqrt{46}}{-\frac{5}{2}}$$

Divide 400-50\sqrt{46} by -\frac{5}{2} by multiplying 400-50\sqrt{46} by the reciprocal of -\frac{5}{2}.

$$x=20\sqrt{46}-160$$

The equation is now solved.

$$x=-20\sqrt{46}-160 x=20\sqrt{46}-160$$

Steps for Completing the Square

Multiply both sides of the equation by 300, the least common multiple of 2,100,3.

$$300\left(10+1+\frac{5}{2}\times \frac{x}{100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Add 10 and 1 to get 11.

$$300\left(11+\frac{5}{2}\times \frac{x}{100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Multiply \frac{5}{2} times \frac{x}{100} by multiplying numerator times numerator and denominator times denominator.

$$300\left(11+\frac{5x}{2\times 100}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Cancel out 5 in both numerator and denominator.

$$300\left(11+\frac{x}{2\times 20}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 11 times \frac{2\times 20}{2\times 20}.

$$300\left(\frac{11\times 2\times 20}{2\times 20}+\frac{x}{2\times 20}\right)\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Since \frac{11\times 2\times 20}{2\times 20} and \frac{x}{2\times 20} have the same denominator, add them by adding their numerators.

$$300\times \frac{11\times 2\times 20+x}{2\times 20}\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Do the multiplications in 11\times 2\times 20+x.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{1}{3}\times \frac{x}{100}\right)-30\right)=75000$$

Multiply \frac{1}{3} times \frac{x}{100} by multiplying numerator times numerator and denominator times denominator.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{x}{3\times 100}\right)-30\right)=75000$$

Multiply 3 and 100 to get 300.

$$300\times \frac{440+x}{2\times 20}\left(50\left(1-\frac{x}{300}\right)-30\right)=75000$$

Use the distributive property to multiply 50 by 1-\frac{x}{300}.

$$300\times \frac{440+x}{2\times 20}\left(50+50\left(-\frac{x}{300}\right)-30\right)=75000$$

Cancel out 300, the greatest common factor in 50 and 300.

$$300\times \frac{440+x}{2\times 20}\left(50+\frac{x}{-6}-30\right)=75000$$

Subtract 30 from 50 to get 20.

$$300\times \frac{440+x}{2\times 20}\left(20+\frac{x}{-6}\right)=75000$$

Express 300\times \frac{440+x}{2\times 20} as a single fraction.

$$\frac{300\left(440+x\right)}{2\times 20}\left(20+\frac{x}{-6}\right)=75000$$

Cancel out 2\times 10 in both numerator and denominator.

$$\frac{15\left(x+440\right)}{2}\left(20+\frac{x}{-6}\right)=75000$$

Use the distributive property to multiply \frac{15\left(x+440\right)}{2} by 20+\frac{x}{-6}.

$$20\times \frac{15\left(x+440\right)}{2}+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 15 by x+440.

$$20\times \frac{15x+6600}{2}+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Cancel out 2, the greatest common factor in 20 and 2.

$$10\left(15x+6600\right)+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 10 by 15x+6600.

$$150x+66000+\frac{15\left(x+440\right)}{2}\times \frac{x}{-6}=75000$$

Use the distributive property to multiply 15 by x+440.

$$150x+66000+\frac{15x+6600}{2}\times \frac{x}{-6}=75000$$

Multiply \frac{15x+6600}{2} times \frac{x}{-6} by multiplying numerator times numerator and denominator times denominator.

$$150x+66000+\frac{\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 150x+66000 times \frac{2\left(-6\right)}{2\left(-6\right)}.

$$\frac{\left(150x+66000\right)\times 2\left(-6\right)}{2\left(-6\right)}+\frac{\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

Since \frac{\left(150x+66000\right)\times 2\left(-6\right)}{2\left(-6\right)} and \frac{\left(15x+6600\right)x}{2\left(-6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(150x+66000\right)\times 2\left(-6\right)+\left(15x+6600\right)x}{2\left(-6\right)}=75000$$

Do the multiplications in \left(150x+66000\right)\times 2\left(-6\right)+\left(15x+6600\right)x.

$$\frac{-1800x-792000+15x^{2}+6600x}{2\left(-6\right)}=75000$$

Combine like terms in -1800x-792000+15x^{2}+6600x.

$$\frac{4800x-792000+15x^{2}}{2\left(-6\right)}=75000$$

Factor the expressions that are not already factored in \frac{4800x-792000+15x^{2}}{2\left(-6\right)}.

$$\frac{15\left(x-120\right)\left(x+440\right)}{-6\times 2}=75000$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\left(x-120\right)\left(x+440\right)}{-2\times 2}=75000$$

Multiply -2 and 2 to get -4.

$$\frac{5\left(x-120\right)\left(x+440\right)}{-4}=75000$$

Use the distributive property to multiply 5 by x-120.

$$\frac{\left(5x-600\right)\left(x+440\right)}{-4}=75000$$

Apply the distributive property by multiplying each term of 5x-600 by each term of x+440.

$$\frac{5x^{2}+2200x-600x-264000}{-4}=75000$$

Combine 2200x and -600x to get 1600x.

$$\frac{5x^{2}+1600x-264000}{-4}=75000$$

Divide each term of 5x^{2}+1600x-264000 by -4 to get -\frac{5}{4}x^{2}-400x+66000.

$$-\frac{5}{4}x^{2}-400x+66000=75000$$

Subtract 66000 from both sides.

$$-\frac{5}{4}x^{2}-400x=75000-66000$$

Subtract 66000 from 75000 to get 9000.

$$-\frac{5}{4}x^{2}-400x=9000$$

Divide both sides of the equation by -\frac{5}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{5}{4}x^{2}-400x}{-\frac{5}{4}}=\frac{9000}{-\frac{5}{4}}$$

Dividing by -\frac{5}{4} undoes the multiplication by -\frac{5}{4}.

$$x^{2}+\left(-\frac{400}{-\frac{5}{4}}\right)x=\frac{9000}{-\frac{5}{4}}$$

Divide -400 by -\frac{5}{4} by multiplying -400 by the reciprocal of -\frac{5}{4}.

$$x^{2}+320x=\frac{9000}{-\frac{5}{4}}$$

Divide 9000 by -\frac{5}{4} by multiplying 9000 by the reciprocal of -\frac{5}{4}.

$$x^{2}+320x=-7200$$

Divide 320, the coefficient of the x term, by 2 to get 160. Then add the square of 160 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+320x+160^{2}=-7200+160^{2}$$

Square 160.

$$x^{2}+320x+25600=-7200+25600$$

Add -7200 to 25600.

$$x^{2}+320x+25600=18400$$

Factor x^{2}+320x+25600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+160\right)^{2}=18400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+160\right)^{2}}=\sqrt{18400}$$

Simplify.

$$x+160=20\sqrt{46} x+160=-20\sqrt{46}$$

Subtract 160 from both sides of the equation.

$$x=20\sqrt{46}-160 x=-20\sqrt{46}-160$$