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

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

Solve for x (complex solution)

$$x=\frac{5\sqrt{150839}i}{39}+\frac{55}{3}\approx 18.333333333+49.792303665i$$

$$x=-\frac{5\sqrt{150839}i}{39}+\frac{55}{3}\approx 18.333333333-49.792303665i$$

Steps Using the Quadratic Formula

Add 30 and 100 to get 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 1000=64000$$

Use the distributive property to multiply 2x-40 by 3x-50 and combine like terms.

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 1000=64000$$

Use the distributive property to multiply 6x^{2}-220x+2000 by 130.

$$780x^{2}-28600x+260000+2000\times 1000=64000$$

Multiply 2000 and 1000 to get 2000000.

$$780x^{2}-28600x+260000+2000000=64000$$

Add 260000 and 2000000 to get 2260000.

$$780x^{2}-28600x+2260000=64000$$

Subtract 64000 from both sides.

$$780x^{2}-28600x+2260000-64000=0$$

Subtract 64000 from 2260000 to get 2196000.

$$780x^{2}-28600x+2196000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 780 for a, -28600 for b, and 2196000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28600\right)±\sqrt{\left(-28600\right)^{2}-4\times 780\times 2196000}}{2\times 780}$$

Square -28600.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-4\times 780\times 2196000}}{2\times 780}$$

Multiply -4 times 780.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-3120\times 2196000}}{2\times 780}$$

Multiply -3120 times 2196000.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-6851520000}}{2\times 780}$$

Add 817960000 to -6851520000.

$$x=\frac{-\left(-28600\right)±\sqrt{-6033560000}}{2\times 780}$$

Take the square root of -6033560000.

$$x=\frac{-\left(-28600\right)±200\sqrt{150839}i}{2\times 780}$$

The opposite of -28600 is 28600.

$$x=\frac{28600±200\sqrt{150839}i}{2\times 780}$$

Multiply 2 times 780.

$$x=\frac{28600±200\sqrt{150839}i}{1560}$$

Now solve the equation x=\frac{28600±200\sqrt{150839}i}{1560} when ± is plus. Add 28600 to 200i\sqrt{150839}.

$$x=\frac{28600+200\sqrt{150839}i}{1560}$$

Divide 28600+200i\sqrt{150839} by 1560.

$$x=\frac{5\sqrt{150839}i}{39}+\frac{55}{3}$$

Now solve the equation x=\frac{28600±200\sqrt{150839}i}{1560} when ± is minus. Subtract 200i\sqrt{150839} from 28600.

$$x=\frac{-200\sqrt{150839}i+28600}{1560}$$

Divide 28600-200i\sqrt{150839} by 1560.

$$x=-\frac{5\sqrt{150839}i}{39}+\frac{55}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{150839}i}{39}+\frac{55}{3} x=-\frac{5\sqrt{150839}i}{39}+\frac{55}{3}$$

Steps for Completing the Square

Add 30 and 100 to get 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 1000=64000$$

Use the distributive property to multiply 2x-40 by 3x-50 and combine like terms.

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 1000=64000$$

Use the distributive property to multiply 6x^{2}-220x+2000 by 130.

$$780x^{2}-28600x+260000+2000\times 1000=64000$$

Multiply 2000 and 1000 to get 2000000.

$$780x^{2}-28600x+260000+2000000=64000$$

Add 260000 and 2000000 to get 2260000.

$$780x^{2}-28600x+2260000=64000$$

Subtract 2260000 from both sides.

$$780x^{2}-28600x=64000-2260000$$

Subtract 2260000 from 64000 to get -2196000.

$$780x^{2}-28600x=-2196000$$

Divide both sides by 780.

$$\frac{780x^{2}-28600x}{780}=-\frac{2196000}{780}$$

Dividing by 780 undoes the multiplication by 780.

$$x^{2}+\left(-\frac{28600}{780}\right)x=-\frac{2196000}{780}$$

Reduce the fraction \frac{-28600}{780} to lowest terms by extracting and canceling out 260.

$$x^{2}-\frac{110}{3}x=-\frac{2196000}{780}$$

Reduce the fraction \frac{-2196000}{780} to lowest terms by extracting and canceling out 60.

$$x^{2}-\frac{110}{3}x=-\frac{36600}{13}$$

Divide -\frac{110}{3}, the coefficient of the x term, by 2 to get -\frac{55}{3}. Then add the square of -\frac{55}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=-\frac{36600}{13}+\left(-\frac{55}{3}\right)^{2}$$

Square -\frac{55}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=-\frac{36600}{13}+\frac{3025}{9}$$

Add -\frac{36600}{13} to \frac{3025}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=-\frac{290075}{117}$$

Factor x^{2}-\frac{110}{3}x+\frac{3025}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{55}{3}\right)^{2}=-\frac{290075}{117}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{-\frac{290075}{117}}$$

Simplify.

$$x-\frac{55}{3}=\frac{5\sqrt{150839}i}{39} x-\frac{55}{3}=-\frac{5\sqrt{150839}i}{39}$$

Add \frac{55}{3} to both sides of the equation.

$$x=\frac{5\sqrt{150839}i}{39}+\frac{55}{3} x=-\frac{5\sqrt{150839}i}{39}+\frac{55}{3}$$