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

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

Solve for x

$$x = \frac{\sqrt{21} + 4}{5} \approx 1.716515139$$

$$x=\frac{4-\sqrt{21}}{5}\approx -0.116515139$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

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

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

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

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

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

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

Add -2 and 3 to get 1.

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

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

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

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

$$x^{2}+1+2x=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$x^{2}+1+2x-6x^{2}=-6x$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+1+2x=-6x$$

Add 6x to both sides.

$$-5x^{2}+1+2x+6x=0$$

Combine 2x and 6x to get 8x.

$$-5x^{2}+1+8x=0$$

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.

$$-5x^{2}+8x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 8 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-5\right)}}{2\left(-5\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-5\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-8±\sqrt{64+20}}{2\left(-5\right)}$$

Add 64 to 20.

$$x=\frac{-8±\sqrt{84}}{2\left(-5\right)}$$

Take the square root of 84.

$$x=\frac{-8±2\sqrt{21}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-8±2\sqrt{21}}{-10}$$

Now solve the equation x=\frac{-8±2\sqrt{21}}{-10} when ± is plus. Add -8 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}-8}{-10}$$

Divide -8+2\sqrt{21} by -10.

$$x=\frac{4-\sqrt{21}}{5}$$

Now solve the equation x=\frac{-8±2\sqrt{21}}{-10} when ± is minus. Subtract 2\sqrt{21} from -8.

$$x=\frac{-2\sqrt{21}-8}{-10}$$

Divide -8-2\sqrt{21} by -10.

$$x=\frac{\sqrt{21}+4}{5}$$

The equation is now solved.

$$x=\frac{4-\sqrt{21}}{5} x=\frac{\sqrt{21}+4}{5}$$

Steps for Completing the Square

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

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

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

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

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

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

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

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

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

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

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

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

Add -2 and 3 to get 1.

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

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

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

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

$$x^{2}+1+2x=6x^{2}-6x$$

Subtract 6x^{2} from both sides.

$$x^{2}+1+2x-6x^{2}=-6x$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+1+2x=-6x$$

Add 6x to both sides.

$$-5x^{2}+1+2x+6x=0$$

Combine 2x and 6x to get 8x.

$$-5x^{2}+1+8x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-5x^{2}+8x=-1$$

Divide both sides by -5.

$$\frac{-5x^{2}+8x}{-5}=-\frac{1}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{8}{-5}x=-\frac{1}{-5}$$

Divide 8 by -5.

$$x^{2}-\frac{8}{5}x=-\frac{1}{-5}$$

Divide -1 by -5.

$$x^{2}-\frac{8}{5}x=\frac{1}{5}$$

Divide -\frac{8}{5}, the coefficient of the x term, by 2 to get -\frac{4}{5}. Then add the square of -\frac{4}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=\frac{1}{5}+\left(-\frac{4}{5}\right)^{2}$$

Square -\frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{1}{5}+\frac{16}{25}$$

Add \frac{1}{5} to \frac{16}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=\frac{21}{25}$$

Factor x^{2}-\frac{8}{5}x+\frac{16}{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}.

$$\left(x-\frac{4}{5}\right)^{2}=\frac{21}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{\frac{21}{25}}$$

Simplify.

$$x-\frac{4}{5}=\frac{\sqrt{21}}{5} x-\frac{4}{5}=-\frac{\sqrt{21}}{5}$$

Add \frac{4}{5} to both sides of the equation.

$$x=\frac{\sqrt{21}+4}{5} x=\frac{4-\sqrt{21}}{5}$$