Solve (100-x/50)((3000+x)-150)-x/50*50=306600

$$(100- \frac{ x }{ 50 } )((3000+x)-150)- \frac{ x }{ 50 } \times 50=306600$$

Solve for x

$$x=900$$

$$x=1200$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 50.

$$50\left(100-\frac{x}{50}\right)\left(3000+x-150\right)-x\times 50=15330000$$

Subtract 150 from 3000 to get 2850.

$$50\left(100-\frac{x}{50}\right)\left(2850+x\right)-x\times 50=15330000$$

Use the distributive property to multiply 50 by 100-\frac{x}{50}.

$$\left(5000+50\left(-\frac{x}{50}\right)\right)\left(2850+x\right)-x\times 50=15330000$$

Express 50\left(-\frac{x}{50}\right) as a single fraction.

$$\left(5000+\frac{-50x}{50}\right)\left(2850+x\right)-x\times 50=15330000$$

Cancel out 50 and 50.

$$\left(5000-x\right)\left(2850+x\right)-x\times 50=15330000$$

Apply the distributive property by multiplying each term of 5000-x by each term of 2850+x.

$$14250000+5000x-2850x-x^{2}-x\times 50=15330000$$

Combine 5000x and -2850x to get 2150x.

$$14250000+2150x-x^{2}-x\times 50=15330000$$

Combine 2150x and -x\times 50 to get 2100x.

$$14250000+2100x-x^{2}=15330000$$

Subtract 15330000 from both sides.

$$14250000+2100x-x^{2}-15330000=0$$

Subtract 15330000 from 14250000 to get -1080000.

$$-1080000+2100x-x^{2}=0$$

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.

$$-x^{2}+2100x-1080000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2100 for b, and -1080000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2100±\sqrt{2100^{2}-4\left(-1\right)\left(-1080000\right)}}{2\left(-1\right)}$$

Square 2100.

$$x=\frac{-2100±\sqrt{4410000-4\left(-1\right)\left(-1080000\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-2100±\sqrt{4410000+4\left(-1080000\right)}}{2\left(-1\right)}$$

Multiply 4 times -1080000.

$$x=\frac{-2100±\sqrt{4410000-4320000}}{2\left(-1\right)}$$

Add 4410000 to -4320000.

$$x=\frac{-2100±\sqrt{90000}}{2\left(-1\right)}$$

Take the square root of 90000.

$$x=\frac{-2100±300}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2100±300}{-2}$$

Now solve the equation x=\frac{-2100±300}{-2} when ± is plus. Add -2100 to 300.

$$x=-\frac{1800}{-2}$$

Divide -1800 by -2.

$$x=900$$

Now solve the equation x=\frac{-2100±300}{-2} when ± is minus. Subtract 300 from -2100.

$$x=-\frac{2400}{-2}$$

Divide -2400 by -2.

$$x=1200$$

The equation is now solved.

$$x=900 x=1200$$

Steps for Completing the Square

Multiply both sides of the equation by 50.

$$50\left(100-\frac{x}{50}\right)\left(3000+x-150\right)-x\times 50=15330000$$

Subtract 150 from 3000 to get 2850.

$$50\left(100-\frac{x}{50}\right)\left(2850+x\right)-x\times 50=15330000$$

Use the distributive property to multiply 50 by 100-\frac{x}{50}.

$$\left(5000+50\left(-\frac{x}{50}\right)\right)\left(2850+x\right)-x\times 50=15330000$$

Express 50\left(-\frac{x}{50}\right) as a single fraction.

$$\left(5000+\frac{-50x}{50}\right)\left(2850+x\right)-x\times 50=15330000$$

Cancel out 50 and 50.

$$\left(5000-x\right)\left(2850+x\right)-x\times 50=15330000$$

Apply the distributive property by multiplying each term of 5000-x by each term of 2850+x.

$$14250000+5000x-2850x-x^{2}-x\times 50=15330000$$

Combine 5000x and -2850x to get 2150x.

$$14250000+2150x-x^{2}-x\times 50=15330000$$

Combine 2150x and -x\times 50 to get 2100x.

$$14250000+2100x-x^{2}=15330000$$

Subtract 14250000 from both sides.

$$2100x-x^{2}=15330000-14250000$$

Subtract 14250000 from 15330000 to get 1080000.

$$2100x-x^{2}=1080000$$

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.

$$-x^{2}+2100x=1080000$$

Divide both sides by -1.

$$\frac{-x^{2}+2100x}{-1}=\frac{1080000}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{2100}{-1}x=\frac{1080000}{-1}$$

Divide 2100 by -1.

$$x^{2}-2100x=\frac{1080000}{-1}$$

Divide 1080000 by -1.

$$x^{2}-2100x=-1080000$$

Divide -2100, the coefficient of the x term, by 2 to get -1050. Then add the square of -1050 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2100x+\left(-1050\right)^{2}=-1080000+\left(-1050\right)^{2}$$

Square -1050.

$$x^{2}-2100x+1102500=-1080000+1102500$$

Add -1080000 to 1102500.

$$x^{2}-2100x+1102500=22500$$

Factor x^{2}-2100x+1102500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1050\right)^{2}=22500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1050\right)^{2}}=\sqrt{22500}$$

Simplify.

$$x-1050=150 x-1050=-150$$

Add 1050 to both sides of the equation.

$$x=1200 x=900$$