Solve 2x/210-x=210-x/2x

$$\frac { 2 x } { 210 - x } = \frac { 210 - x } { 2 x }$$

Solve for x

$$x=-210$$

$$x=70$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,210 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-210\right), the least common multiple of 210-x,2x.

$$-2x\times 2x=\left(x-210\right)\left(210-x\right)$$

Multiply -2 and 2 to get -4.

$$-4xx=\left(x-210\right)\left(210-x\right)$$

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

$$-4x^{2}=\left(x-210\right)\left(210-x\right)$$

Use the distributive property to multiply x-210 by 210-x and combine like terms.

$$-4x^{2}=420x-x^{2}-44100$$

Subtract 420x from both sides.

$$-4x^{2}-420x=-x^{2}-44100$$

Add x^{2} to both sides.

$$-4x^{2}-420x+x^{2}=-44100$$

Combine -4x^{2} and x^{2} to get -3x^{2}.

$$-3x^{2}-420x=-44100$$

Add 44100 to both sides.

$$-3x^{2}-420x+44100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -420 for b, and 44100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-420\right)±\sqrt{\left(-420\right)^{2}-4\left(-3\right)\times 44100}}{2\left(-3\right)}$$

Square -420.

$$x=\frac{-\left(-420\right)±\sqrt{176400-4\left(-3\right)\times 44100}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-420\right)±\sqrt{176400+12\times 44100}}{2\left(-3\right)}$$

Multiply 12 times 44100.

$$x=\frac{-\left(-420\right)±\sqrt{176400+529200}}{2\left(-3\right)}$$

Add 176400 to 529200.

$$x=\frac{-\left(-420\right)±\sqrt{705600}}{2\left(-3\right)}$$

Take the square root of 705600.

$$x=\frac{-\left(-420\right)±840}{2\left(-3\right)}$$

The opposite of -420 is 420.

$$x=\frac{420±840}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{420±840}{-6}$$

Now solve the equation x=\frac{420±840}{-6} when ± is plus. Add 420 to 840.

$$x=\frac{1260}{-6}$$

Divide 1260 by -6.

$$x=-210$$

Now solve the equation x=\frac{420±840}{-6} when ± is minus. Subtract 840 from 420.

$$x=-\frac{420}{-6}$$

Divide -420 by -6.

$$x=70$$

The equation is now solved.

$$x=-210 x=70$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,210 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-210\right), the least common multiple of 210-x,2x.

$$-2x\times 2x=\left(x-210\right)\left(210-x\right)$$

Multiply -2 and 2 to get -4.

$$-4xx=\left(x-210\right)\left(210-x\right)$$

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

$$-4x^{2}=\left(x-210\right)\left(210-x\right)$$

Use the distributive property to multiply x-210 by 210-x and combine like terms.

$$-4x^{2}=420x-x^{2}-44100$$

Subtract 420x from both sides.

$$-4x^{2}-420x=-x^{2}-44100$$

Add x^{2} to both sides.

$$-4x^{2}-420x+x^{2}=-44100$$

Combine -4x^{2} and x^{2} to get -3x^{2}.

$$-3x^{2}-420x=-44100$$

Divide both sides by -3.

$$\frac{-3x^{2}-420x}{-3}=-\frac{44100}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{420}{-3}\right)x=-\frac{44100}{-3}$$

Divide -420 by -3.

$$x^{2}+140x=-\frac{44100}{-3}$$

Divide -44100 by -3.

$$x^{2}+140x=14700$$

Divide 140, the coefficient of the x term, by 2 to get 70. Then add the square of 70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+140x+70^{2}=14700+70^{2}$$

Square 70.

$$x^{2}+140x+4900=14700+4900$$

Add 14700 to 4900.

$$x^{2}+140x+4900=19600$$

Factor x^{2}+140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+70\right)^{2}=19600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+70\right)^{2}}=\sqrt{19600}$$

Simplify.

$$x+70=140 x+70=-140$$

Subtract 70 from both sides of the equation.

$$x=70 x=-210$$