Solve ((3x-50)(2x-40)+(2x-40)(40))*30+2x*3x*100=642000

$$((3x-50)(2x-40)+(2x-40)(40)) \times 30+2x \times 3x \times 100=642000$$

Solve for x

$$x = \frac{5 \sqrt{5509} + 35}{13} \approx 31.239476175$$

$$x=\frac{35-5\sqrt{5509}}{13}\approx -25.854860791$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$\left(\left(3x-50\right)\left(2x-40\right)+\left(2x-40\right)\times 40\right)\times 30+2x^{2}\times 3\times 100=642000$$

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

$$\left(6x^{2}-220x+2000+\left(2x-40\right)\times 40\right)\times 30+2x^{2}\times 3\times 100=642000$$

Use the distributive property to multiply 2x-40 by 40.

$$\left(6x^{2}-220x+2000+80x-1600\right)\times 30+2x^{2}\times 3\times 100=642000$$

Combine -220x and 80x to get -140x.

$$\left(6x^{2}-140x+2000-1600\right)\times 30+2x^{2}\times 3\times 100=642000$$

Subtract 1600 from 2000 to get 400.

$$\left(6x^{2}-140x+400\right)\times 30+2x^{2}\times 3\times 100=642000$$

Use the distributive property to multiply 6x^{2}-140x+400 by 30.

$$180x^{2}-4200x+12000+2x^{2}\times 3\times 100=642000$$

Multiply 2 and 3 to get 6.

$$180x^{2}-4200x+12000+6x^{2}\times 100=642000$$

Multiply 6 and 100 to get 600.

$$180x^{2}-4200x+12000+600x^{2}=642000$$

Combine 180x^{2} and 600x^{2} to get 780x^{2}.

$$780x^{2}-4200x+12000=642000$$

Subtract 642000 from both sides.

$$780x^{2}-4200x+12000-642000=0$$

Subtract 642000 from 12000 to get -630000.

$$780x^{2}-4200x-630000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 780 for a, -4200 for b, and -630000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4200\right)±\sqrt{\left(-4200\right)^{2}-4\times 780\left(-630000\right)}}{2\times 780}$$

Square -4200.

$$x=\frac{-\left(-4200\right)±\sqrt{17640000-4\times 780\left(-630000\right)}}{2\times 780}$$

Multiply -4 times 780.

$$x=\frac{-\left(-4200\right)±\sqrt{17640000-3120\left(-630000\right)}}{2\times 780}$$

Multiply -3120 times -630000.

$$x=\frac{-\left(-4200\right)±\sqrt{17640000+1965600000}}{2\times 780}$$

Add 17640000 to 1965600000.

$$x=\frac{-\left(-4200\right)±\sqrt{1983240000}}{2\times 780}$$

Take the square root of 1983240000.

$$x=\frac{-\left(-4200\right)±600\sqrt{5509}}{2\times 780}$$

The opposite of -4200 is 4200.

$$x=\frac{4200±600\sqrt{5509}}{2\times 780}$$

Multiply 2 times 780.

$$x=\frac{4200±600\sqrt{5509}}{1560}$$

Now solve the equation x=\frac{4200±600\sqrt{5509}}{1560} when ± is plus. Add 4200 to 600\sqrt{5509}.

$$x=\frac{600\sqrt{5509}+4200}{1560}$$

Divide 4200+600\sqrt{5509} by 1560.

$$x=\frac{5\sqrt{5509}+35}{13}$$

Now solve the equation x=\frac{4200±600\sqrt{5509}}{1560} when ± is minus. Subtract 600\sqrt{5509} from 4200.

$$x=\frac{4200-600\sqrt{5509}}{1560}$$

Divide 4200-600\sqrt{5509} by 1560.

$$x=\frac{35-5\sqrt{5509}}{13}$$

The equation is now solved.

$$x=\frac{5\sqrt{5509}+35}{13} x=\frac{35-5\sqrt{5509}}{13}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$\left(\left(3x-50\right)\left(2x-40\right)+\left(2x-40\right)\times 40\right)\times 30+2x^{2}\times 3\times 100=642000$$

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

$$\left(6x^{2}-220x+2000+\left(2x-40\right)\times 40\right)\times 30+2x^{2}\times 3\times 100=642000$$

Use the distributive property to multiply 2x-40 by 40.

$$\left(6x^{2}-220x+2000+80x-1600\right)\times 30+2x^{2}\times 3\times 100=642000$$

Combine -220x and 80x to get -140x.

$$\left(6x^{2}-140x+2000-1600\right)\times 30+2x^{2}\times 3\times 100=642000$$

Subtract 1600 from 2000 to get 400.

$$\left(6x^{2}-140x+400\right)\times 30+2x^{2}\times 3\times 100=642000$$

Use the distributive property to multiply 6x^{2}-140x+400 by 30.

$$180x^{2}-4200x+12000+2x^{2}\times 3\times 100=642000$$

Multiply 2 and 3 to get 6.

$$180x^{2}-4200x+12000+6x^{2}\times 100=642000$$

Multiply 6 and 100 to get 600.

$$180x^{2}-4200x+12000+600x^{2}=642000$$

Combine 180x^{2} and 600x^{2} to get 780x^{2}.

$$780x^{2}-4200x+12000=642000$$

Subtract 12000 from both sides.

$$780x^{2}-4200x=642000-12000$$

Subtract 12000 from 642000 to get 630000.

$$780x^{2}-4200x=630000$$

Divide both sides by 780.

$$\frac{780x^{2}-4200x}{780}=\frac{630000}{780}$$

Dividing by 780 undoes the multiplication by 780.

$$x^{2}+\left(-\frac{4200}{780}\right)x=\frac{630000}{780}$$

Reduce the fraction \frac{-4200}{780} to lowest terms by extracting and canceling out 60.

$$x^{2}-\frac{70}{13}x=\frac{630000}{780}$$

Reduce the fraction \frac{630000}{780} to lowest terms by extracting and canceling out 60.

$$x^{2}-\frac{70}{13}x=\frac{10500}{13}$$

Divide -\frac{70}{13}, the coefficient of the x term, by 2 to get -\frac{35}{13}. Then add the square of -\frac{35}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{70}{13}x+\left(-\frac{35}{13}\right)^{2}=\frac{10500}{13}+\left(-\frac{35}{13}\right)^{2}$$

Square -\frac{35}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{70}{13}x+\frac{1225}{169}=\frac{10500}{13}+\frac{1225}{169}$$

Add \frac{10500}{13} to \frac{1225}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{70}{13}x+\frac{1225}{169}=\frac{137725}{169}$$

Factor x^{2}-\frac{70}{13}x+\frac{1225}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{13}\right)^{2}=\frac{137725}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{13}\right)^{2}}=\sqrt{\frac{137725}{169}}$$

Simplify.

$$x-\frac{35}{13}=\frac{5\sqrt{5509}}{13} x-\frac{35}{13}=-\frac{5\sqrt{5509}}{13}$$

Add \frac{35}{13} to both sides of the equation.

$$x=\frac{5\sqrt{5509}+35}{13} x=\frac{35-5\sqrt{5509}}{13}$$