Solve x+1/x^2-6x+9-x+3/x^2-4x+3=2/x^2-2x+1

$$\frac { x + 1 } { x ^ { 2 } - 6 x + 9 } - \frac { x + 3 } { x ^ { 2 } - 4 x + 3 } = \frac { 2 } { x ^ { 2 } - 2 x + 1 }$$

Solve for x

$$x=2\sqrt{3}+5\approx 8.464101615$$

$$x=5-2\sqrt{3}\approx 1.535898385$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)^{2}\left(x-1\right)^{2}, the least common multiple of x^{2}-6x+9,x^{2}-4x+3,x^{2}-2x+1.

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

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

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

Use the distributive property to multiply x^{2}-2x+1 by x+1 and combine like terms.

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

Use the distributive property to multiply x-3 by x-1 and combine like terms.

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

Use the distributive property to multiply x^{2}-4x+3 by x+3 and combine like terms.

$$x^{3}-x^{2}-x+1-\left(x^{3}-x^{2}-9x+9\right)=\left(x-3\right)^{2}\times 2$$

To find the opposite of x^{3}-x^{2}-9x+9, find the opposite of each term.

$$x^{3}-x^{2}-x+1-x^{3}+x^{2}+9x-9=\left(x-3\right)^{2}\times 2$$

Combine x^{3} and -x^{3} to get 0.

$$-x^{2}-x+1+x^{2}+9x-9=\left(x-3\right)^{2}\times 2$$

Combine -x^{2} and x^{2} to get 0.

$$-x+1+9x-9=\left(x-3\right)^{2}\times 2$$

Combine -x and 9x to get 8x.

$$8x+1-9=\left(x-3\right)^{2}\times 2$$

Subtract 9 from 1 to get -8.

$$8x-8=\left(x-3\right)^{2}\times 2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$8x-8=\left(x^{2}-6x+9\right)\times 2$$

Use the distributive property to multiply x^{2}-6x+9 by 2.

$$8x-8=2x^{2}-12x+18$$

Subtract 2x^{2} from both sides.

$$8x-8-2x^{2}=-12x+18$$

Add 12x to both sides.

$$8x-8-2x^{2}+12x=18$$

Combine 8x and 12x to get 20x.

$$20x-8-2x^{2}=18$$

Subtract 18 from both sides.

$$20x-8-2x^{2}-18=0$$

Subtract 18 from -8 to get -26.

$$20x-26-2x^{2}=0$$

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.

$$-2x^{2}+20x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 20 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-2\right)\left(-26\right)}}{2\left(-2\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-2\right)\left(-26\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-20±\sqrt{400+8\left(-26\right)}}{2\left(-2\right)}$$

Multiply 8 times -26.

$$x=\frac{-20±\sqrt{400-208}}{2\left(-2\right)}$$

Add 400 to -208.

$$x=\frac{-20±\sqrt{192}}{2\left(-2\right)}$$

Take the square root of 192.

$$x=\frac{-20±8\sqrt{3}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-20±8\sqrt{3}}{-4}$$

Now solve the equation x=\frac{-20±8\sqrt{3}}{-4} when ± is plus. Add -20 to 8\sqrt{3}.

$$x=\frac{8\sqrt{3}-20}{-4}$$

Divide -20+8\sqrt{3} by -4.

$$x=5-2\sqrt{3}$$

Now solve the equation x=\frac{-20±8\sqrt{3}}{-4} when ± is minus. Subtract 8\sqrt{3} from -20.

$$x=\frac{-8\sqrt{3}-20}{-4}$$

Divide -20-8\sqrt{3} by -4.

$$x=2\sqrt{3}+5$$

The equation is now solved.

$$x=5-2\sqrt{3} x=2\sqrt{3}+5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)^{2}\left(x-1\right)^{2}, the least common multiple of x^{2}-6x+9,x^{2}-4x+3,x^{2}-2x+1.

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

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

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

Use the distributive property to multiply x^{2}-2x+1 by x+1 and combine like terms.

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

Use the distributive property to multiply x-3 by x-1 and combine like terms.

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

Use the distributive property to multiply x^{2}-4x+3 by x+3 and combine like terms.

$$x^{3}-x^{2}-x+1-\left(x^{3}-x^{2}-9x+9\right)=\left(x-3\right)^{2}\times 2$$

To find the opposite of x^{3}-x^{2}-9x+9, find the opposite of each term.

$$x^{3}-x^{2}-x+1-x^{3}+x^{2}+9x-9=\left(x-3\right)^{2}\times 2$$

Combine x^{3} and -x^{3} to get 0.

$$-x^{2}-x+1+x^{2}+9x-9=\left(x-3\right)^{2}\times 2$$

Combine -x^{2} and x^{2} to get 0.

$$-x+1+9x-9=\left(x-3\right)^{2}\times 2$$

Combine -x and 9x to get 8x.

$$8x+1-9=\left(x-3\right)^{2}\times 2$$

Subtract 9 from 1 to get -8.

$$8x-8=\left(x-3\right)^{2}\times 2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$8x-8=\left(x^{2}-6x+9\right)\times 2$$

Use the distributive property to multiply x^{2}-6x+9 by 2.

$$8x-8=2x^{2}-12x+18$$

Subtract 2x^{2} from both sides.

$$8x-8-2x^{2}=-12x+18$$

Add 12x to both sides.

$$8x-8-2x^{2}+12x=18$$

Combine 8x and 12x to get 20x.

$$20x-8-2x^{2}=18$$

Add 8 to both sides.

$$20x-2x^{2}=18+8$$

Add 18 and 8 to get 26.

$$20x-2x^{2}=26$$

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.

$$-2x^{2}+20x=26$$

Divide both sides by -2.

$$\frac{-2x^{2}+20x}{-2}=\frac{26}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{20}{-2}x=\frac{26}{-2}$$

Divide 20 by -2.

$$x^{2}-10x=\frac{26}{-2}$$

Divide 26 by -2.

$$x^{2}-10x=-13$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-13+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-13+25$$

Add -13 to 25.

$$x^{2}-10x+25=12$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{12}$$

Simplify.

$$x-5=2\sqrt{3} x-5=-2\sqrt{3}$$

Add 5 to both sides of the equation.

$$x=2\sqrt{3}+5 x=5-2\sqrt{3}$$