Solve 1/(x-1)^2+3/x-1-10=0.1

$$\frac { 1 } { ( x - 1 ) ^ { 2 } } + \frac { 3 } { x - 1 } - 10 = 0.1$$

Solve for x

$$x=\frac{116-\sqrt{1235}}{101}\approx 0.800568646$$

$$x = \frac{\sqrt{1235} + 116}{101} \approx 1.496461057$$

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 \left(x-1\right)^{2}, the least common multiple of \left(x-1\right)^{2},x-1.

$$1+\left(x-1\right)\times 3+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use the distributive property to multiply x-1 by 3.

$$1+3x-3+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Subtract 3 from 1 to get -2.

$$-2+3x+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-2+3x+\left(x^{2}-2x+1\right)\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use the distributive property to multiply x^{2}-2x+1 by -10.

$$-2+3x-10x^{2}+20x-10=0.1\left(x-1\right)^{2}$$

Combine 3x and 20x to get 23x.

$$-2+23x-10x^{2}-10=0.1\left(x-1\right)^{2}$$

Subtract 10 from -2 to get -12.

$$-12+23x-10x^{2}=0.1\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-12+23x-10x^{2}=0.1\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply 0.1 by x^{2}-2x+1.

$$-12+23x-10x^{2}=0.1x^{2}-0.2x+0.1$$

Subtract 0.1x^{2} from both sides.

$$-12+23x-10x^{2}-0.1x^{2}=-0.2x+0.1$$

Combine -10x^{2} and -0.1x^{2} to get -10.1x^{2}.

$$-12+23x-10.1x^{2}=-0.2x+0.1$$

Add 0.2x to both sides.

$$-12+23x-10.1x^{2}+0.2x=0.1$$

Combine 23x and 0.2x to get 23.2x.

$$-12+23.2x-10.1x^{2}=0.1$$

Subtract 0.1 from both sides.

$$-12+23.2x-10.1x^{2}-0.1=0$$

Subtract 0.1 from -12 to get -12.1.

$$-12.1+23.2x-10.1x^{2}=0$$

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.

$$-10.1x^{2}+23.2x-12.1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10.1 for a, 23.2 for b, and -12.1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23.2±\sqrt{23.2^{2}-4\left(-10.1\right)\left(-12.1\right)}}{2\left(-10.1\right)}$$

Square 23.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-23.2±\sqrt{538.24-4\left(-10.1\right)\left(-12.1\right)}}{2\left(-10.1\right)}$$

Multiply -4 times -10.1.

$$x=\frac{-23.2±\sqrt{538.24+40.4\left(-12.1\right)}}{2\left(-10.1\right)}$$

Multiply 40.4 times -12.1 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-23.2±\sqrt{\frac{13456-12221}{25}}}{2\left(-10.1\right)}$$

Add 538.24 to -488.84 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-23.2±\sqrt{49.4}}{2\left(-10.1\right)}$$

Take the square root of 49.4.

$$x=\frac{-23.2±\frac{\sqrt{1235}}{5}}{2\left(-10.1\right)}$$

Multiply 2 times -10.1.

$$x=\frac{-23.2±\frac{\sqrt{1235}}{5}}{-20.2}$$

Now solve the equation x=\frac{-23.2±\frac{\sqrt{1235}}{5}}{-20.2} when ± is plus. Add -23.2 to \frac{\sqrt{1235}}{5}.

$$x=\frac{\sqrt{1235}-116}{-20.2\times 5}$$

Divide \frac{-116+\sqrt{1235}}{5} by -20.2 by multiplying \frac{-116+\sqrt{1235}}{5} by the reciprocal of -20.2.

$$x=\frac{116-\sqrt{1235}}{101}$$

Now solve the equation x=\frac{-23.2±\frac{\sqrt{1235}}{5}}{-20.2} when ± is minus. Subtract \frac{\sqrt{1235}}{5} from -23.2.

$$x=\frac{-\sqrt{1235}-116}{-20.2\times 5}$$

Divide \frac{-116-\sqrt{1235}}{5} by -20.2 by multiplying \frac{-116-\sqrt{1235}}{5} by the reciprocal of -20.2.

$$x=\frac{\sqrt{1235}+116}{101}$$

The equation is now solved.

$$x=\frac{116-\sqrt{1235}}{101} x=\frac{\sqrt{1235}+116}{101}$$

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 \left(x-1\right)^{2}, the least common multiple of \left(x-1\right)^{2},x-1.

$$1+\left(x-1\right)\times 3+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use the distributive property to multiply x-1 by 3.

$$1+3x-3+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Subtract 3 from 1 to get -2.

$$-2+3x+\left(x-1\right)^{2}\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-2+3x+\left(x^{2}-2x+1\right)\left(-10\right)=0.1\left(x-1\right)^{2}$$

Use the distributive property to multiply x^{2}-2x+1 by -10.

$$-2+3x-10x^{2}+20x-10=0.1\left(x-1\right)^{2}$$

Combine 3x and 20x to get 23x.

$$-2+23x-10x^{2}-10=0.1\left(x-1\right)^{2}$$

Subtract 10 from -2 to get -12.

$$-12+23x-10x^{2}=0.1\left(x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-12+23x-10x^{2}=0.1\left(x^{2}-2x+1\right)$$

Use the distributive property to multiply 0.1 by x^{2}-2x+1.

$$-12+23x-10x^{2}=0.1x^{2}-0.2x+0.1$$

Subtract 0.1x^{2} from both sides.

$$-12+23x-10x^{2}-0.1x^{2}=-0.2x+0.1$$

Combine -10x^{2} and -0.1x^{2} to get -10.1x^{2}.

$$-12+23x-10.1x^{2}=-0.2x+0.1$$

Add 0.2x to both sides.

$$-12+23x-10.1x^{2}+0.2x=0.1$$

Combine 23x and 0.2x to get 23.2x.

$$-12+23.2x-10.1x^{2}=0.1$$

Add 12 to both sides.

$$23.2x-10.1x^{2}=0.1+12$$

Add 0.1 and 12 to get 12.1.

$$23.2x-10.1x^{2}=12.1$$

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.

$$-10.1x^{2}+23.2x=12.1$$

Divide both sides of the equation by -10.1, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-10.1x^{2}+23.2x}{-10.1}=\frac{12.1}{-10.1}$$

Dividing by -10.1 undoes the multiplication by -10.1.

$$x^{2}+\frac{23.2}{-10.1}x=\frac{12.1}{-10.1}$$

Divide 23.2 by -10.1 by multiplying 23.2 by the reciprocal of -10.1.

$$x^{2}-\frac{232}{101}x=\frac{12.1}{-10.1}$$

Divide 12.1 by -10.1 by multiplying 12.1 by the reciprocal of -10.1.

$$x^{2}-\frac{232}{101}x=-\frac{121}{101}$$

Divide -\frac{232}{101}, the coefficient of the x term, by 2 to get -\frac{116}{101}. Then add the square of -\frac{116}{101} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{232}{101}x+\left(-\frac{116}{101}\right)^{2}=-\frac{121}{101}+\left(-\frac{116}{101}\right)^{2}$$

Square -\frac{116}{101} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{232}{101}x+\frac{13456}{10201}=-\frac{121}{101}+\frac{13456}{10201}$$

Add -\frac{121}{101} to \frac{13456}{10201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{232}{101}x+\frac{13456}{10201}=\frac{1235}{10201}$$

Factor x^{2}-\frac{232}{101}x+\frac{13456}{10201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{116}{101}\right)^{2}=\frac{1235}{10201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{116}{101}\right)^{2}}=\sqrt{\frac{1235}{10201}}$$

Simplify.

$$x-\frac{116}{101}=\frac{\sqrt{1235}}{101} x-\frac{116}{101}=-\frac{\sqrt{1235}}{101}$$

Add \frac{116}{101} to both sides of the equation.

$$x=\frac{\sqrt{1235}+116}{101} x=\frac{116-\sqrt{1235}}{101}$$