Solve (2x-3)(x-40*50)*(30+100)+2000*1000=64000

$$( 2 x - 3 ) ( x - 40 \times 50 ) \times ( 30 + 100 ) + 2000 \times 1000 = 64000$$

Solve for x

$$x=\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}\approx 1996.267156383$$

$$x=-\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}\approx 5.232843617$$

Steps Using the Quadratic Formula

Multiply 40 and 50 to get 2000.

$$\left(2x-3\right)\left(x-2000\right)\left(30+100\right)+2000\times 1000=64000$$

Add 30 and 100 to get 130.

$$\left(2x-3\right)\left(x-2000\right)\times 130+2000\times 1000=64000$$

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

$$\left(2x^{2}-4003x+6000\right)\times 130+2000\times 1000=64000$$

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

$$260x^{2}-520390x+780000+2000\times 1000=64000$$

Multiply 2000 and 1000 to get 2000000.

$$260x^{2}-520390x+780000+2000000=64000$$

Add 780000 and 2000000 to get 2780000.

$$260x^{2}-520390x+2780000=64000$$

Subtract 64000 from both sides.

$$260x^{2}-520390x+2780000-64000=0$$

Subtract 64000 from 2780000 to get 2716000.

$$260x^{2}-520390x+2716000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 260 for a, -520390 for b, and 2716000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-520390\right)±\sqrt{\left(-520390\right)^{2}-4\times 260\times 2716000}}{2\times 260}$$

Square -520390.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-4\times 260\times 2716000}}{2\times 260}$$

Multiply -4 times 260.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-1040\times 2716000}}{2\times 260}$$

Multiply -1040 times 2716000.

$$x=\frac{-\left(-520390\right)±\sqrt{270805752100-2824640000}}{2\times 260}$$

Add 270805752100 to -2824640000.

$$x=\frac{-\left(-520390\right)±\sqrt{267981112100}}{2\times 260}$$

Take the square root of 267981112100.

$$x=\frac{-\left(-520390\right)±10\sqrt{2679811121}}{2\times 260}$$

The opposite of -520390 is 520390.

$$x=\frac{520390±10\sqrt{2679811121}}{2\times 260}$$

Multiply 2 times 260.

$$x=\frac{520390±10\sqrt{2679811121}}{520}$$

Now solve the equation x=\frac{520390±10\sqrt{2679811121}}{520} when ± is plus. Add 520390 to 10\sqrt{2679811121}.

$$x=\frac{10\sqrt{2679811121}+520390}{520}$$

Divide 520390+10\sqrt{2679811121} by 520.

$$x=\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}$$

Now solve the equation x=\frac{520390±10\sqrt{2679811121}}{520} when ± is minus. Subtract 10\sqrt{2679811121} from 520390.

$$x=\frac{520390-10\sqrt{2679811121}}{520}$$

Divide 520390-10\sqrt{2679811121} by 520.

$$x=-\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{2679811121}}{52}+\frac{4003}{4} x=-\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}$$

Steps for Completing the Square

Multiply 40 and 50 to get 2000.

$$\left(2x-3\right)\left(x-2000\right)\left(30+100\right)+2000\times 1000=64000$$

Add 30 and 100 to get 130.

$$\left(2x-3\right)\left(x-2000\right)\times 130+2000\times 1000=64000$$

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

$$\left(2x^{2}-4003x+6000\right)\times 130+2000\times 1000=64000$$

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

$$260x^{2}-520390x+780000+2000\times 1000=64000$$

Multiply 2000 and 1000 to get 2000000.

$$260x^{2}-520390x+780000+2000000=64000$$

Add 780000 and 2000000 to get 2780000.

$$260x^{2}-520390x+2780000=64000$$

Subtract 2780000 from both sides.

$$260x^{2}-520390x=64000-2780000$$

Subtract 2780000 from 64000 to get -2716000.

$$260x^{2}-520390x=-2716000$$

Divide both sides by 260.

$$\frac{260x^{2}-520390x}{260}=-\frac{2716000}{260}$$

Dividing by 260 undoes the multiplication by 260.

$$x^{2}+\left(-\frac{520390}{260}\right)x=-\frac{2716000}{260}$$

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

$$x^{2}-\frac{4003}{2}x=-\frac{2716000}{260}$$

Reduce the fraction \frac{-2716000}{260} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{4003}{2}x=-\frac{135800}{13}$$

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.

$$x^{2}-\frac{4003}{2}x+\left(-\frac{4003}{4}\right)^{2}=-\frac{135800}{13}+\left(-\frac{4003}{4}\right)^{2}$$

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

$$x^{2}-\frac{4003}{2}x+\frac{16024009}{16}=-\frac{135800}{13}+\frac{16024009}{16}$$

Add -\frac{135800}{13} to \frac{16024009}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4003}{2}x+\frac{16024009}{16}=\frac{206139317}{208}$$

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}.

$$\left(x-\frac{4003}{4}\right)^{2}=\frac{206139317}{208}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4003}{4}\right)^{2}}=\sqrt{\frac{206139317}{208}}$$

Simplify.

$$x-\frac{4003}{4}=\frac{\sqrt{2679811121}}{52} x-\frac{4003}{4}=-\frac{\sqrt{2679811121}}{52}$$

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

$$x=\frac{\sqrt{2679811121}}{52}+\frac{4003}{4} x=-\frac{\sqrt{2679811121}}{52}+\frac{4003}{4}$$