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

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

Solve for x

$$x=-4$$

$$x=-3$$

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 3\left(x+1\right), the least common multiple of x+1,3.

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

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

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

Multiply 3 and -2 to get -6.

$$3x-3-6\left(x+1\right)=\left(x+1\right)\left(x+3\right)$$

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

$$3x-3-6x-6=\left(x+1\right)\left(x+3\right)$$

Combine 3x and -6x to get -3x.

$$-3x-3-6=\left(x+1\right)\left(x+3\right)$$

Subtract 6 from -3 to get -9.

$$-3x-9=\left(x+1\right)\left(x+3\right)$$

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

$$-3x-9=x^{2}+4x+3$$

Subtract x^{2} from both sides.

$$-3x-9-x^{2}=4x+3$$

Subtract 4x from both sides.

$$-3x-9-x^{2}-4x=3$$

Combine -3x and -4x to get -7x.

$$-7x-9-x^{2}=3$$

Subtract 3 from both sides.

$$-7x-9-x^{2}-3=0$$

Subtract 3 from -9 to get -12.

$$-7x-12-x^{2}=0$$

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.

$$-x^{2}-7x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -7 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-7\right)±\sqrt{49+4\left(-12\right)}}{2\left(-1\right)}$$

Multiply 4 times -12.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2\left(-1\right)}$$

Add 49 to -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$x=\frac{-\left(-7\right)±1}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{7±1}{-2}$$

Now solve the equation x=\frac{7±1}{-2} when ± is plus. Add 7 to 1.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

Now solve the equation x=\frac{7±1}{-2} when ± is minus. Subtract 1 from 7.

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

Divide 6 by -2.

$$x=-3$$

The equation is now solved.

$$x=-4 x=-3$$

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 3\left(x+1\right), the least common multiple of x+1,3.

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

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

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

Multiply 3 and -2 to get -6.

$$3x-3-6\left(x+1\right)=\left(x+1\right)\left(x+3\right)$$

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

$$3x-3-6x-6=\left(x+1\right)\left(x+3\right)$$

Combine 3x and -6x to get -3x.

$$-3x-3-6=\left(x+1\right)\left(x+3\right)$$

Subtract 6 from -3 to get -9.

$$-3x-9=\left(x+1\right)\left(x+3\right)$$

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

$$-3x-9=x^{2}+4x+3$$

Subtract x^{2} from both sides.

$$-3x-9-x^{2}=4x+3$$

Subtract 4x from both sides.

$$-3x-9-x^{2}-4x=3$$

Combine -3x and -4x to get -7x.

$$-7x-9-x^{2}=3$$

Add 9 to both sides.

$$-7x-x^{2}=3+9$$

Add 3 and 9 to get 12.

$$-7x-x^{2}=12$$

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.

$$-x^{2}-7x=12$$

Divide both sides by -1.

$$\frac{-x^{2}-7x}{-1}=\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{7}{-1}\right)x=\frac{12}{-1}$$

Divide -7 by -1.

$$x^{2}+7x=\frac{12}{-1}$$

Divide 12 by -1.

$$x^{2}+7x=-12$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-12+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=-12+\frac{49}{4}$$

Add -12 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{1}{4}$$

Factor x^{2}+7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{1}{2} x+\frac{7}{2}=-\frac{1}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=-3 x=-4$$