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

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

Solve for x

$$x=18$$

$$x=24$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 100-x by 3000+50x and combine like terms.

$$300000+2000x-50x^{2}-150\left(100-x\right)-50x=306600$$

Use the distributive property to multiply -150 by 100-x.

$$300000+2000x-50x^{2}-15000+150x-50x=306600$$

Subtract 15000 from 300000 to get 285000.

$$285000+2000x-50x^{2}+150x-50x=306600$$

Combine 2000x and 150x to get 2150x.

$$285000+2150x-50x^{2}-50x=306600$$

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

$$285000+2100x-50x^{2}=306600$$

Subtract 306600 from both sides.

$$285000+2100x-50x^{2}-306600=0$$

Subtract 306600 from 285000 to get -21600.

$$-21600+2100x-50x^{2}=0$$

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.

$$-50x^{2}+2100x-21600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -50 for a, 2100 for b, and -21600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2100±\sqrt{2100^{2}-4\left(-50\right)\left(-21600\right)}}{2\left(-50\right)}$$

Square 2100.

$$x=\frac{-2100±\sqrt{4410000-4\left(-50\right)\left(-21600\right)}}{2\left(-50\right)}$$

Multiply -4 times -50.

$$x=\frac{-2100±\sqrt{4410000+200\left(-21600\right)}}{2\left(-50\right)}$$

Multiply 200 times -21600.

$$x=\frac{-2100±\sqrt{4410000-4320000}}{2\left(-50\right)}$$

Add 4410000 to -4320000.

$$x=\frac{-2100±\sqrt{90000}}{2\left(-50\right)}$$

Take the square root of 90000.

$$x=\frac{-2100±300}{2\left(-50\right)}$$

Multiply 2 times -50.

$$x=\frac{-2100±300}{-100}$$

Now solve the equation x=\frac{-2100±300}{-100} when ± is plus. Add -2100 to 300.

$$x=-\frac{1800}{-100}$$

Divide -1800 by -100.

$$x=18$$

Now solve the equation x=\frac{-2100±300}{-100} when ± is minus. Subtract 300 from -2100.

$$x=-\frac{2400}{-100}$$

Divide -2400 by -100.

$$x=24$$

The equation is now solved.

$$x=18 x=24$$

Steps for Completing the Square

Use the distributive property to multiply 100-x by 3000+50x and combine like terms.

$$300000+2000x-50x^{2}-150\left(100-x\right)-50x=306600$$

Use the distributive property to multiply -150 by 100-x.

$$300000+2000x-50x^{2}-15000+150x-50x=306600$$

Subtract 15000 from 300000 to get 285000.

$$285000+2000x-50x^{2}+150x-50x=306600$$

Combine 2000x and 150x to get 2150x.

$$285000+2150x-50x^{2}-50x=306600$$

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

$$285000+2100x-50x^{2}=306600$$

Subtract 285000 from both sides.

$$2100x-50x^{2}=306600-285000$$

Subtract 285000 from 306600 to get 21600.

$$2100x-50x^{2}=21600$$

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.

$$-50x^{2}+2100x=21600$$

Divide both sides by -50.

$$\frac{-50x^{2}+2100x}{-50}=\frac{21600}{-50}$$

Dividing by -50 undoes the multiplication by -50.

$$x^{2}+\frac{2100}{-50}x=\frac{21600}{-50}$$

Divide 2100 by -50.

$$x^{2}-42x=\frac{21600}{-50}$$

Divide 21600 by -50.

$$x^{2}-42x=-432$$

Divide -42, the coefficient of the x term, by 2 to get -21. Then add the square of -21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-42x+\left(-21\right)^{2}=-432+\left(-21\right)^{2}$$

Square -21.

$$x^{2}-42x+441=-432+441$$

Add -432 to 441.

$$x^{2}-42x+441=9$$

Factor x^{2}-42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-21\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-21\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-21=3 x-21=-3$$

Add 21 to both sides of the equation.

$$x=24 x=18$$