Solve 6/2-x/2001-1=1-x/2002-x/2003

$$6 / \frac { 2 - x } { 2001 } - 1 = \frac { 1 - x } { 2002 } - \frac { x } { 2003 }$$

Solve for x

$$x = \frac{\sqrt{787301003208721} + 4020019}{8010} \approx 4004.856874925$$

$$x=\frac{4020019-\sqrt{787301003208721}}{8010}\approx -3001.106812503$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4010006, the least common multiple of 2002,2003.

$$8024022006\left(2-x\right)^{-1}\times 6-4010006=2003\left(1-x\right)-2002x$$

Multiply 8024022006 and 6 to get 48144132036.

$$48144132036\left(2-x\right)^{-1}-4010006=2003\left(1-x\right)-2002x$$

Use the distributive property to multiply 2003 by 1-x.

$$48144132036\left(2-x\right)^{-1}-4010006=2003-2003x-2002x$$

Combine -2003x and -2002x to get -4005x.

$$48144132036\left(2-x\right)^{-1}-4010006=2003-4005x$$

Subtract 2003 from both sides.

$$48144132036\left(2-x\right)^{-1}-4010006-2003=-4005x$$

Subtract 2003 from -4010006 to get -4012009.

$$48144132036\left(2-x\right)^{-1}-4012009=-4005x$$

Add 4005x to both sides.

$$48144132036\left(2-x\right)^{-1}-4012009+4005x=0$$

Reorder the terms.

$$4005x-4012009+48144132036\times \frac{1}{-x+2}=0$$

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

$$4005x\left(-x+2\right)+\left(-x+2\right)\left(-4012009\right)[phone]\times 1=0$$

Use the distributive property to multiply 4005x by -x+2.

$$-4005x^{2}+8010x+\left(-x+2\right)\left(-4012009\right)[phone]\times 1=0$$

Use the distributive property to multiply -x+2 by -4012009.

$$-4005x^{2}+8010x+4012009x-8024018+48144132036\times 1=0$$

Combine 8010x and 4012009x to get 4020019x.

$$-4005x^{2}+4020019x-8024018+48144132036\times 1=0$$

Multiply 48144132036 and 1 to get 48144132036.

$$-4005x^{2}+4020019x-8024018+48144132036=0$$

Add -8024018 and 48144132036 to get 48136108018.

$$-4005x^{2}+4020019x+48136108018=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4005 for a, 4020019 for b, and 48136108018 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4020019±\sqrt{4020019^{2}-4\left(-4005\right)\times 48136108018}}{2\left(-4005\right)}$$

Square 4020019.

$$x=\frac{-4020019±\sqrt{16160552760361-4\left(-4005\right)\times 48136108018}}{2\left(-4005\right)}$$

Multiply -4 times -4005.

$$x=\frac{-4020019±\sqrt{16160552760361+16020\times 48136108018}}{2\left(-4005\right)}$$

Multiply 16020 times 48136108018.

$$x=\frac{-4020019±\sqrt{16160552760361+771140450448360}}{2\left(-4005\right)}$$

Add 16160552760361 to 771140450448360.

$$x=\frac{-4020019±\sqrt{787301003208721}}{2\left(-4005\right)}$$

Multiply 2 times -4005.

$$x=\frac{-4020019±\sqrt{787301003208721}}{-8010}$$

Now solve the equation x=\frac{-4020019±\sqrt{787301003208721}}{-8010} when ± is plus. Add -4020019 to \sqrt{787301003208721}.

$$x=\frac{\sqrt{787301003208721}-4020019}{-8010}$$

Divide -4020019+\sqrt{787301003208721} by -8010.

$$x=\frac{4020019-\sqrt{787301003208721}}{8010}$$

Now solve the equation x=\frac{-4020019±\sqrt{787301003208721}}{-8010} when ± is minus. Subtract \sqrt{787301003208721} from -4020019.

$$x=\frac{-\sqrt{787301003208721}-4020019}{-8010}$$

Divide -4020019-\sqrt{787301003208721} by -8010.

$$x=\frac{\sqrt{787301003208721}+4020019}{8010}$$

The equation is now solved.

$$x=\frac{4020019-\sqrt{787301003208721}}{8010} x=\frac{\sqrt{787301003208721}+4020019}{8010}$$

Steps for Completing the Square

Multiply both sides of the equation by 4010006, the least common multiple of 2002,2003.

$$8024022006\left(2-x\right)^{-1}\times 6-4010006=2003\left(1-x\right)-2002x$$

Multiply 8024022006 and 6 to get 48144132036.

$$48144132036\left(2-x\right)^{-1}-4010006=2003\left(1-x\right)-2002x$$

Use the distributive property to multiply 2003 by 1-x.

$$48144132036\left(2-x\right)^{-1}-4010006=2003-2003x-2002x$$

Combine -2003x and -2002x to get -4005x.

$$48144132036\left(2-x\right)^{-1}-4010006=2003-4005x$$

Add 4005x to both sides.

$$48144132036\left(2-x\right)^{-1}-4010006+4005x=2003$$

Add 4010006 to both sides.

$$48144132036\left(2-x\right)^{-1}+4005x=2003+4010006$$

Add 2003 and 4010006 to get 4012009.

$$48144132036\left(2-x\right)^{-1}+4005x=4012009$$

Reorder the terms.

$$4005x+48144132036\times \frac{1}{-x+2}=4012009$$

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by -x+2.

$$4005x\left(-x+2\right)[phone]\times 1=4012009\left(-x+2\right)$$

Use the distributive property to multiply 4005x by -x+2.

$$-4005x^{2}+8010x+48144132036\times 1=4012009\left(-x+2\right)$$

Multiply 48144132036 and 1 to get 48144132036.

$$-4005x^{2}+8010x+48144132036=4012009\left(-x+2\right)$$

Use the distributive property to multiply 4012009 by -x+2.

$$-4005x^{2}+8010x+48144132036=-4012009x+8024018$$

Add 4012009x to both sides.

$$-4005x^{2}+8010x+48144132036+4012009x=8024018$$

Combine 8010x and 4012009x to get 4020019x.

$$-4005x^{2}+4020019x+48144132036=8024018$$

Subtract 48144132036 from both sides.

$$-4005x^{2}+4020019x=8024018-48144132036$$

Subtract 48144132036 from 8024018 to get -48136108018.

$$-4005x^{2}+4020019x=-48136108018$$

Divide both sides by -4005.

$$\frac{-4005x^{2}+4020019x}{-4005}=-\frac{48136108018}{-4005}$$

Dividing by -4005 undoes the multiplication by -4005.

$$x^{2}+\frac{4020019}{-4005}x=-\frac{48136108018}{-4005}$$

Divide 4020019 by -4005.

$$x^{2}-\frac{4020019}{4005}x=-\frac{48136108018}{-4005}$$

Divide -48136108018 by -4005.

$$x^{2}-\frac{4020019}{4005}x=\frac{48136108018}{4005}$$

Divide -\frac{4020019}{4005}, the coefficient of the x term, by 2 to get -\frac{4020019}{8010}. Then add the square of -\frac{4020019}{8010} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4020019}{4005}x+\left(-\frac{4020019}{8010}\right)^{2}=\frac{48136108018}{4005}+\left(-\frac{4020019}{8010}\right)^{2}$$

Square -\frac{4020019}{8010} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4020019}{4005}x+\frac{16160552760361}{64160100}=\frac{48136108018}{4005}+\frac{16160552760361}{64160100}$$

Add \frac{48136108018}{4005} to \frac{16160552760361}{64160100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4020019}{4005}x+\frac{16160552760361}{64160100}=\frac{787301003208721}{64160100}$$

Factor x^{2}-\frac{4020019}{4005}x+\frac{16160552760361}{64160100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4020019}{8010}\right)^{2}=\frac{787301003208721}{64160100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4020019}{8010}\right)^{2}}=\sqrt{\frac{787301003208721}{64160100}}$$

Simplify.

$$x-\frac{4020019}{8010}=\frac{\sqrt{787301003208721}}{8010} x-\frac{4020019}{8010}=-\frac{\sqrt{787301003208721}}{8010}$$

Add \frac{4020019}{8010} to both sides of the equation.

$$x=\frac{\sqrt{787301003208721}+4020019}{8010} x=\frac{4020019-\sqrt{787301003208721}}{8010}$$