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

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

Solve for x

$$x=-2$$

$$x=1$$

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 2\left(x+1\right), the least common multiple of x+1,2.

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

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

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

Multiply -\frac{1}{2} and 2 to get -1.

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

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

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 4 by x+1.

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

Combine -x and 4x to get 3x.

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

Add x^{2} to both sides.

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

Subtract 3x from both sides.

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

Combine 4x and -3x to get x.

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

Subtract 4 from both sides.

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

Subtract 4 from 2 to get -2.

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

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.

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

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

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

Square 1.

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

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8}}{2}$$

Add 1 to 8.

$$x=\frac{-1±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-1±3}{2}$$

Now solve the equation x=\frac{-1±3}{2} when ± is plus. Add -1 to 3.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-1±3}{2} when ± is minus. Subtract 3 from -1.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=1 x=-2$$

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 2\left(x+1\right), the least common multiple of x+1,2.

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

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

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

Multiply -\frac{1}{2} and 2 to get -1.

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

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

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

Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply 4 by x+1.

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

Combine -x and 4x to get 3x.

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

Add x^{2} to both sides.

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

Subtract 3x from both sides.

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

Combine 4x and -3x to get x.

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

Subtract 2 from both sides.

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

Subtract 2 from 4 to get 2.

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

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.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 2 to \frac{1}{4}.

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

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Subtract \frac{1}{2} from both sides of the equation.

$$x=1 x=-2$$