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

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

Solve for x

$$x = \frac{5 \sqrt{205} + 55}{3} \approx 42.196368439$$

$$x=\frac{55-5\sqrt{205}}{3}\approx -5.529701772$$

Steps Using the Quadratic Formula

Add 30 and 100 to get 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 100=642000$$

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

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 100=642000$$

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

$$780x^{2}-28600x+260000+2000\times 100=642000$$

Multiply 2000 and 100 to get 200000.

$$780x^{2}-28600x+260000+200000=642000$$

Add 260000 and 200000 to get 460000.

$$780x^{2}-28600x+460000=642000$$

Subtract 642000 from both sides.

$$780x^{2}-28600x+460000-642000=0$$

Subtract 642000 from 460000 to get -182000.

$$780x^{2}-28600x-182000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 780 for a, -28600 for b, and -182000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28600\right)±\sqrt{\left(-28600\right)^{2}-4\times 780\left(-182000\right)}}{2\times 780}$$

Square -28600.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-4\times 780\left(-182000\right)}}{2\times 780}$$

Multiply -4 times 780.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000-3120\left(-182000\right)}}{2\times 780}$$

Multiply -3120 times -182000.

$$x=\frac{-\left(-28600\right)±\sqrt{817960000+567840000}}{2\times 780}$$

Add 817960000 to 567840000.

$$x=\frac{-\left(-28600\right)±\sqrt{1385800000}}{2\times 780}$$

Take the square root of 1385800000.

$$x=\frac{-\left(-28600\right)±2600\sqrt{205}}{2\times 780}$$

The opposite of -28600 is 28600.

$$x=\frac{28600±2600\sqrt{205}}{2\times 780}$$

Multiply 2 times 780.

$$x=\frac{28600±2600\sqrt{205}}{1560}$$

Now solve the equation x=\frac{28600±2600\sqrt{205}}{1560} when ± is plus. Add 28600 to 2600\sqrt{205}.

$$x=\frac{2600\sqrt{205}+28600}{1560}$$

Divide 28600+2600\sqrt{205} by 1560.

$$x=\frac{5\sqrt{205}+55}{3}$$

Now solve the equation x=\frac{28600±2600\sqrt{205}}{1560} when ± is minus. Subtract 2600\sqrt{205} from 28600.

$$x=\frac{28600-2600\sqrt{205}}{1560}$$

Divide 28600-2600\sqrt{205} by 1560.

$$x=\frac{55-5\sqrt{205}}{3}$$

The equation is now solved.

$$x=\frac{5\sqrt{205}+55}{3} x=\frac{55-5\sqrt{205}}{3}$$

Steps for Completing the Square

Add 30 and 100 to get 130.

$$\left(2x-40\right)\left(3x-50\right)\times 130+2000\times 100=642000$$

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

$$\left(6x^{2}-220x+2000\right)\times 130+2000\times 100=642000$$

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

$$780x^{2}-28600x+260000+2000\times 100=642000$$

Multiply 2000 and 100 to get 200000.

$$780x^{2}-28600x+260000+200000=642000$$

Add 260000 and 200000 to get 460000.

$$780x^{2}-28600x+460000=642000$$

Subtract 460000 from both sides.

$$780x^{2}-28600x=642000-460000$$

Subtract 460000 from 642000 to get 182000.

$$780x^{2}-28600x=182000$$

Divide both sides by 780.

$$\frac{780x^{2}-28600x}{780}=\frac{182000}{780}$$

Dividing by 780 undoes the multiplication by 780.

$$x^{2}+\left(-\frac{28600}{780}\right)x=\frac{182000}{780}$$

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

$$x^{2}-\frac{110}{3}x=\frac{182000}{780}$$

Reduce the fraction \frac{182000}{780} to lowest terms by extracting and canceling out 260.

$$x^{2}-\frac{110}{3}x=\frac{700}{3}$$

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.

$$x^{2}-\frac{110}{3}x+\left(-\frac{55}{3}\right)^{2}=\frac{700}{3}+\left(-\frac{55}{3}\right)^{2}$$

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

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{700}{3}+\frac{3025}{9}$$

Add \frac{700}{3} to \frac{3025}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{110}{3}x+\frac{3025}{9}=\frac{5125}{9}$$

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}.

$$\left(x-\frac{55}{3}\right)^{2}=\frac{5125}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{55}{3}\right)^{2}}=\sqrt{\frac{5125}{9}}$$

Simplify.

$$x-\frac{55}{3}=\frac{5\sqrt{205}}{3} x-\frac{55}{3}=-\frac{5\sqrt{205}}{3}$$

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

$$x=\frac{5\sqrt{205}+55}{3} x=\frac{55-5\sqrt{205}}{3}$$