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

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

Solve for x

$$x=\sqrt{7}+3\approx 5.645751311$$

$$x=3-\sqrt{7}\approx 0.354248689$$

Steps Using the Quadratic Formula

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

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

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

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

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

$$2x+2+3x-6=\left(x-2\right)\left(x+1\right)$$

Combine 2x and 3x to get 5x.

$$5x+2-6=\left(x-2\right)\left(x+1\right)$$

Subtract 6 from 2 to get -4.

$$5x-4=\left(x-2\right)\left(x+1\right)$$

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

$$5x-4=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$5x-4-x^{2}=-x-2$$

Add x to both sides.

$$5x-4-x^{2}+x=-2$$

Combine 5x and x to get 6x.

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

Add 2 to both sides.

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

Add -4 and 2 to get -2.

$$6x-2-x^{2}=0$$

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$x=\frac{-6±\sqrt{36-8}}{2\left(-1\right)}$$

Add 36 to -8.

$$x=\frac{-6±\sqrt{28}}{2\left(-1\right)}$$

Take the square root of 28.

$$x=\frac{-6±2\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±2\sqrt{7}}{-2}$$

Now solve the equation x=\frac{-6±2\sqrt{7}}{-2} when ± is plus. Add -6 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-6}{-2}$$

Divide -6+2\sqrt{7} by -2.

$$x=3-\sqrt{7}$$

Now solve the equation x=\frac{-6±2\sqrt{7}}{-2} when ± is minus. Subtract 2\sqrt{7} from -6.

$$x=\frac{-2\sqrt{7}-6}{-2}$$

Divide -6-2\sqrt{7} by -2.

$$x=\sqrt{7}+3$$

The equation is now solved.

$$x=3-\sqrt{7} x=\sqrt{7}+3$$

Steps for Completing the Square

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

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

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

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

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

$$2x+2+3x-6=\left(x-2\right)\left(x+1\right)$$

Combine 2x and 3x to get 5x.

$$5x+2-6=\left(x-2\right)\left(x+1\right)$$

Subtract 6 from 2 to get -4.

$$5x-4=\left(x-2\right)\left(x+1\right)$$

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

$$5x-4=x^{2}-x-2$$

Subtract x^{2} from both sides.

$$5x-4-x^{2}=-x-2$$

Add x to both sides.

$$5x-4-x^{2}+x=-2$$

Combine 5x and x to get 6x.

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

Add 4 to both sides.

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

Add -2 and 4 to get 2.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 6 by -1.

$$x^{2}-6x=\frac{2}{-1}$$

Divide 2 by -1.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -3.

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

Add -2 to 9.

$$x^{2}-6x+9=7$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-3=\sqrt{7} x-3=-\sqrt{7}$$

Add 3 to both sides of the equation.

$$x=\sqrt{7}+3 x=3-\sqrt{7}$$