Solve x-1/x-3-2/x^2+3x=1

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

Solve for x (complex solution)

$$x=-1+\sqrt{2}i\approx -1+1.414213562i$$

$$x=-\sqrt{2}i-1\approx -1-1.414213562i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}+3x.

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

Use the distributive property to multiply x by x+3.

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

Use the distributive property to multiply x^{2}+3x by x-1 and combine like terms.

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

Use the distributive property to multiply x-3 by 2.

$$x^{3}+2x^{2}-3x-\left(2x-6\right)=x\left(x-3\right)\left(x+3\right)$$

To find the opposite of 2x-6, find the opposite of each term.

$$x^{3}+2x^{2}-3x-2x+6=x\left(x-3\right)\left(x+3\right)$$

Combine -3x and -2x to get -5x.

$$x^{3}+2x^{2}-5x+6=x\left(x-3\right)\left(x+3\right)$$

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

$$x^{3}+2x^{2}-5x+6=\left(x^{2}-3x\right)\left(x+3\right)$$

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

$$x^{3}+2x^{2}-5x+6=x^{3}-9x$$

Subtract x^{3} from both sides.

$$x^{3}+2x^{2}-5x+6-x^{3}=-9x$$

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

$$2x^{2}-5x+6=-9x$$

Add 9x to both sides.

$$2x^{2}-5x+6+9x=0$$

Combine -5x and 9x to get 4x.

$$2x^{2}+4x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\times 6}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\times 6}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\times 6}}{2\times 2}$$

Multiply -8 times 6.

$$x=\frac{-4±\sqrt{16-48}}{2\times 2}$$

Add 16 to -48.

$$x=\frac{-4±\sqrt{-32}}{2\times 2}$$

Take the square root of -32.

$$x=\frac{-4±4\sqrt{2}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{2}i}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{2}i}{4} when ± is plus. Add -4 to 4i\sqrt{2}.

$$x=\frac{-4+2^{\frac{5}{2}}i}{4}$$

Divide -4+i\times 2^{\frac{5}{2}} by 4.

$$x=-1+\sqrt{2}i$$

Now solve the equation x=\frac{-4±4\sqrt{2}i}{4} when ± is minus. Subtract 4i\sqrt{2} from -4.

$$x=\frac{-2^{\frac{5}{2}}i-4}{4}$$

Divide -4-i\times 2^{\frac{5}{2}} by 4.

$$x=-\sqrt{2}i-1$$

The equation is now solved.

$$x=-1+\sqrt{2}i x=-\sqrt{2}i-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right)\left(x+3\right), the least common multiple of x-3,x^{2}+3x.

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

Use the distributive property to multiply x by x+3.

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

Use the distributive property to multiply x^{2}+3x by x-1 and combine like terms.

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

Use the distributive property to multiply x-3 by 2.

$$x^{3}+2x^{2}-3x-\left(2x-6\right)=x\left(x-3\right)\left(x+3\right)$$

To find the opposite of 2x-6, find the opposite of each term.

$$x^{3}+2x^{2}-3x-2x+6=x\left(x-3\right)\left(x+3\right)$$

Combine -3x and -2x to get -5x.

$$x^{3}+2x^{2}-5x+6=x\left(x-3\right)\left(x+3\right)$$

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

$$x^{3}+2x^{2}-5x+6=\left(x^{2}-3x\right)\left(x+3\right)$$

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

$$x^{3}+2x^{2}-5x+6=x^{3}-9x$$

Subtract x^{3} from both sides.

$$x^{3}+2x^{2}-5x+6-x^{3}=-9x$$

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

$$2x^{2}-5x+6=-9x$$

Add 9x to both sides.

$$2x^{2}-5x+6+9x=0$$

Combine -5x and 9x to get 4x.

$$2x^{2}+4x+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}+4x=-6$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 4 by 2.

$$x^{2}+2x=-\frac{6}{2}$$

Divide -6 by 2.

$$x^{2}+2x=-3$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=-3+1^{2}$$

Square 1.

$$x^{2}+2x+1=-3+1$$

Add -3 to 1.

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

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2}i x+1=-\sqrt{2}i$$

Subtract 1 from both sides of the equation.

$$x=-1+\sqrt{2}i x=-\sqrt{2}i-1$$