Solve left(2x-3right)left(x-40*50right)left(30+100right)+2000*1000=642000

$$\left( 2x-3 \right) \left( x-40 \cdot 50 \right) \left( 30+100 \right) +2000 \cdot 1000 = 642000$$

Solve for x

$$x=\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}\approx 1997.383074695$$

$$x=-\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}\approx 4.116925305$$

Steps Using the Quadratic Formula

Multiply 40 and 50 to get 2000.

$$\left(2x-3\right)\left(x-2000\right)\left(30+100\right)+2000\times 1000=642000$$

Add 30 and 100 to get 130.

$$\left(2x-3\right)\left(x-2000\right)\times 130+2000\times 1000=642000$$

Use the distributive property to multiply 2x-3 by x-2000 and combine like terms.

$$\left(2x^{2}-4003x+6000\right)\times 130+2000\times 1000=642000$$

Use the distributive property to multiply 2x^{2}-4003x+6000 by 130.

$$260x^{2}-520390x+780000+2000\times 1000=642000$$

Multiply 2000 and 1000 to get 2000000.

$$260x^{2}-520390x+780000+2000000=642000$$

Add 780000 and 2000000 to get 2780000.

$$260x^{2}-520390x+2780000=642000$$

Subtract 642000 from both sides.

$$260x^{2}-520390x+2780000-642000=0$$

Subtract 642000 from 2780000 to get 2138000.

$$260x^{2}-520390x+2138000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 260 for a, -520390 for b, and 2138000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-520390\right)±\sqrt{\left(-520390\right)^{2}-4\times 260\times 2138000}}{2\times 260}$$

Square -520390.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-4\times 260\times 2138000}}{2\times 260}$$

Multiply -4 times 260.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-1040\times 2138000}}{2\times 260}$$

Multiply -1040 times 2138000.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-2223520000}}{2\times 260}$$

Add 270805752100 to -2223520000.

$$x=\frac{-\left(-520390\right)±\sqrt{268582232100}}{2\times 260}$$

Take the square root of 268582232100.

$$x=\frac{-\left(-520390\right)±10\sqrt{2685822321}}{2\times 260}$$

The opposite of -520390 is 520390.

$$x=\frac{520390±10\sqrt{2685822321}}{2\times 260}$$

Multiply 2 times 260.

$$x=\frac{520390±10\sqrt{2685822321}}{520}$$

Now solve the equation x=\frac{520390±10\sqrt{2685822321}}{520} when ± is plus. Add 520390 to 10\sqrt{2685822321}.

$$x=\frac{10\sqrt{2685822321}+520390}{520}$$

Divide 520390+10\sqrt{2685822321} by 520.

$$x=\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}$$

Now solve the equation x=\frac{520390±10\sqrt{2685822321}}{520} when ± is minus. Subtract 10\sqrt{2685822321} from 520390.

$$x=\frac{520390-10\sqrt{2685822321}}{520}$$

Divide 520390-10\sqrt{2685822321} by 520.

$$x=-\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{2685822321}}{52}+\frac{4003}{4} x=-\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}$$

Steps for Completing the Square

Multiply 40 and 50 to get 2000.

$$\left(2x-3\right)\left(x-2000\right)\left(30+100\right)+2000\times 1000=642000$$

Add 30 and 100 to get 130.

$$\left(2x-3\right)\left(x-2000\right)\times 130+2000\times 1000=642000$$

Use the distributive property to multiply 2x-3 by x-2000 and combine like terms.

$$\left(2x^{2}-4003x+6000\right)\times 130+2000\times 1000=642000$$

Use the distributive property to multiply 2x^{2}-4003x+6000 by 130.

$$260x^{2}-520390x+780000+2000\times 1000=642000$$

Multiply 2000 and 1000 to get 2000000.

$$260x^{2}-520390x+780000+2000000=642000$$

Add 780000 and 2000000 to get 2780000.

$$260x^{2}-520390x+2780000=642000$$

Subtract 2780000 from both sides.

$$260x^{2}-520390x=642000-2780000$$

Subtract 2780000 from 642000 to get -2138000.

$$260x^{2}-520390x=-2138000$$

Divide both sides by 260.

$$\frac{260x^{2}-520390x}{260}=-\frac{2138000}{260}$$

Dividing by 260 undoes the multiplication by 260.

$$x^{2}+\left(-\frac{520390}{260}\right)x=-\frac{2138000}{260}$$

Reduce the fraction \frac{-520390}{260} to lowest terms by extracting and canceling out 130.

$$x^{2}-\frac{4003}{2}x=-\frac{2138000}{260}$$

Reduce the fraction \frac{-2138000}{260} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{4003}{2}x=-\frac{106900}{13}$$

Divide -\frac{4003}{2}, the coefficient of the x term, by 2 to get -\frac{4003}{4}. Then add the square of -\frac{4003}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4003}{2}x+\left(-\frac{4003}{4}\right)^{2}=-\frac{106900}{13}+\left(-\frac{4003}{4}\right)^{2}$$

Square -\frac{4003}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4003}{2}x+\frac{16024009}{16}=-\frac{106900}{13}+\frac{16024009}{16}$$

Add -\frac{106900}{13} to \frac{16024009}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4003}{2}x+\frac{16024009}{16}=\frac{206601717}{208}$$

Factor x^{2}-\frac{4003}{2}x+\frac{16024009}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4003}{4}\right)^{2}=\frac{206601717}{208}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4003}{4}\right)^{2}}=\sqrt{\frac{206601717}{208}}$$

Simplify.

$$x-\frac{4003}{4}=\frac{\sqrt{2685822321}}{52} x-\frac{4003}{4}=-\frac{\sqrt{2685822321}}{52}$$

Add \frac{4003}{4} to both sides of the equation.

$$x=\frac{\sqrt{2685822321}}{52}+\frac{4003}{4} x=-\frac{\sqrt{2685822321}}{52}+\frac{4003}{4}$$