Solve 1/(1-x)2=1+2x+3x+4x

$$\frac{ 1 }{ (1-x)2 } =1+2x+3x+4x$$

Solve for x

$$x=\frac{\sqrt{82}}{18}+\frac{4}{9}\approx 0.947521397$$

$$x=-\frac{\sqrt{82}}{18}+\frac{4}{9}\approx -0.058632508$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(-x+1\right).

$$1=2\left(-x+1\right)+2x\times 2\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 2 by -x+1.

$$1=-2x+2+2x\times 2\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Multiply 2 and 2 to get 4.

$$1=-2x+2+4x\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 4x by -x+1.

$$1=-2x+2-4x^{2}+4x+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Combine -2x and 4x to get 2x.

$$1=2x+2-4x^{2}+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Multiply 3 and 2 to get 6.

$$1=2x+2-4x^{2}+6x\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 6x by -x+1.

$$1=2x+2-4x^{2}-6x^{2}+6x+4x\times 2\left(-x+1\right)$$

Combine -4x^{2} and -6x^{2} to get -10x^{2}.

$$1=2x+2-10x^{2}+6x+4x\times 2\left(-x+1\right)$$

Combine 2x and 6x to get 8x.

$$1=8x+2-10x^{2}+4x\times 2\left(-x+1\right)$$

Multiply 4 and 2 to get 8.

$$1=8x+2-10x^{2}+8x\left(-x+1\right)$$

Use the distributive property to multiply 8x by -x+1.

$$1=8x+2-10x^{2}-8x^{2}+8x$$

Combine -10x^{2} and -8x^{2} to get -18x^{2}.

$$1=8x+2-18x^{2}+8x$$

Combine 8x and 8x to get 16x.

$$1=16x+2-18x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$16x+2-18x^{2}=1$$

Subtract 1 from both sides.

$$16x+2-18x^{2}-1=0$$

Subtract 1 from 2 to get 1.

$$16x+1-18x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-18x^{2}+16x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 16 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-18\right)}}{2\left(-18\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-18\right)}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-16±\sqrt{256+72}}{2\left(-18\right)}$$

Add 256 to 72.

$$x=\frac{-16±\sqrt{328}}{2\left(-18\right)}$$

Take the square root of 328.

$$x=\frac{-16±2\sqrt{82}}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-16±2\sqrt{82}}{-36}$$

Now solve the equation x=\frac{-16±2\sqrt{82}}{-36} when ± is plus. Add -16 to 2\sqrt{82}.

$$x=\frac{2\sqrt{82}-16}{-36}$$

Divide -16+2\sqrt{82} by -36.

$$x=-\frac{\sqrt{82}}{18}+\frac{4}{9}$$

Now solve the equation x=\frac{-16±2\sqrt{82}}{-36} when ± is minus. Subtract 2\sqrt{82} from -16.

$$x=\frac{-2\sqrt{82}-16}{-36}$$

Divide -16-2\sqrt{82} by -36.

$$x=\frac{\sqrt{82}}{18}+\frac{4}{9}$$

The equation is now solved.

$$x=-\frac{\sqrt{82}}{18}+\frac{4}{9} x=\frac{\sqrt{82}}{18}+\frac{4}{9}$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 2\left(-x+1\right).

$$1=2\left(-x+1\right)+2x\times 2\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 2 by -x+1.

$$1=-2x+2+2x\times 2\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Multiply 2 and 2 to get 4.

$$1=-2x+2+4x\left(-x+1\right)+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 4x by -x+1.

$$1=-2x+2-4x^{2}+4x+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Combine -2x and 4x to get 2x.

$$1=2x+2-4x^{2}+3x\times 2\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Multiply 3 and 2 to get 6.

$$1=2x+2-4x^{2}+6x\left(-x+1\right)+4x\times 2\left(-x+1\right)$$

Use the distributive property to multiply 6x by -x+1.

$$1=2x+2-4x^{2}-6x^{2}+6x+4x\times 2\left(-x+1\right)$$

Combine -4x^{2} and -6x^{2} to get -10x^{2}.

$$1=2x+2-10x^{2}+6x+4x\times 2\left(-x+1\right)$$

Combine 2x and 6x to get 8x.

$$1=8x+2-10x^{2}+4x\times 2\left(-x+1\right)$$

Multiply 4 and 2 to get 8.

$$1=8x+2-10x^{2}+8x\left(-x+1\right)$$

Use the distributive property to multiply 8x by -x+1.

$$1=8x+2-10x^{2}-8x^{2}+8x$$

Combine -10x^{2} and -8x^{2} to get -18x^{2}.

$$1=8x+2-18x^{2}+8x$$

Combine 8x and 8x to get 16x.

$$1=16x+2-18x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$16x+2-18x^{2}=1$$

Subtract 2 from both sides.

$$16x-18x^{2}=1-2$$

Subtract 2 from 1 to get -1.

$$16x-18x^{2}=-1$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-18x^{2}+16x=-1$$

Divide both sides by -18.

$$\frac{-18x^{2}+16x}{-18}=-\frac{1}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{16}{-18}x=-\frac{1}{-18}$$

Reduce the fraction \frac{16}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{9}x=-\frac{1}{-18}$$

Divide -1 by -18.

$$x^{2}-\frac{8}{9}x=\frac{1}{18}$$

Divide -\frac{8}{9}, the coefficient of the x term, by 2 to get -\frac{4}{9}. Then add the square of -\frac{4}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{9}x+\left(-\frac{4}{9}\right)^{2}=\frac{1}{18}+\left(-\frac{4}{9}\right)^{2}$$

Square -\frac{4}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{1}{18}+\frac{16}{81}$$

Add \frac{1}{18} to \frac{16}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{41}{162}$$

Factor x^{2}-\frac{8}{9}x+\frac{16}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{9}\right)^{2}=\frac{41}{162}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{9}\right)^{2}}=\sqrt{\frac{41}{162}}$$

Simplify.

$$x-\frac{4}{9}=\frac{\sqrt{82}}{18} x-\frac{4}{9}=-\frac{\sqrt{82}}{18}$$

Add \frac{4}{9} to both sides of the equation.

$$x=\frac{\sqrt{82}}{18}+\frac{4}{9} x=-\frac{\sqrt{82}}{18}+\frac{4}{9}$$