Solve 3x+7/x+2+x-1/x=2

$$\frac { 3 x + 7 } { x + 2 } + \frac { x - 1 } { x } = 2$$

Solve for x (complex solution)

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

Steps Using the Quadratic Formula

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

$$x\left(3x+7\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x by 3x+7.

$$3x^{2}+7x+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

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

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Combine 7x and x to get 8x.

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

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

$$4x^{2}+8x-2=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x^{2}+8x-2-2x^{2}=4x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}+8x-2=4x$$

Subtract 4x from both sides.

$$2x^{2}+8x-2-4x=0$$

Combine 8x and -4x to get 4x.

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

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-2\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-2\right)}}{2\times 2}$$

Multiply -8 times -2.

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

Add 16 to 16.

$$x=\frac{-4±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

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

Multiply 2 times 2.

$$x=\frac{-4±2^{\frac{5}{2}}}{4}$$

Now solve the equation x=\frac{-4±2^{\frac{5}{2}}}{4} when ± is plus. Add -4 to 2^{\frac{5}{2}}.

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

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

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

Now solve the equation x=\frac{-4±2^{\frac{5}{2}}}{4} when ± is minus. Subtract 2^{\frac{5}{2}} from -4.

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

Divide -4-4\sqrt{2} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$x\left(3x+7\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x by 3x+7.

$$3x^{2}+7x+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

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

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Combine 7x and x to get 8x.

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

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

$$4x^{2}+8x-2=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x^{2}+8x-2-2x^{2}=4x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}+8x-2=4x$$

Subtract 4x from both sides.

$$2x^{2}+8x-2-4x=0$$

Combine 8x and -4x to get 4x.

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

Add 2 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 4 by 2.

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

Divide 2 by 2.

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

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.

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

Square 1.

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

Add 1 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.

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

Subtract 1 from both sides of the equation.

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

Solve for x

$$x=\sqrt{2}-1\approx 0.414213562$$

$$x=-\sqrt{2}-1\approx -2.414213562$$

Steps Using the Quadratic Formula

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

$$x\left(3x+7\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x by 3x+7.

$$3x^{2}+7x+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

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

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Combine 7x and x to get 8x.

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

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

$$4x^{2}+8x-2=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x^{2}+8x-2-2x^{2}=4x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}+8x-2=4x$$

Subtract 4x from both sides.

$$2x^{2}+8x-2-4x=0$$

Combine 8x and -4x to get 4x.

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

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-2\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-2\right)}}{2\times 2}$$

Multiply -8 times -2.

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

Add 16 to 16.

$$x=\frac{-4±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

$$x=\frac{-4±4\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{2}}{4}$$

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

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

Divide -4+4\sqrt{2} by 4.

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

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

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

Divide -4-4\sqrt{2} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$x\left(3x+7\right)+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x by 3x+7.

$$3x^{2}+7x+\left(x+2\right)\left(x-1\right)=2x\left(x+2\right)$$

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

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

Combine 3x^{2} and x^{2} to get 4x^{2}.

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

Combine 7x and x to get 8x.

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

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

$$4x^{2}+8x-2=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x^{2}+8x-2-2x^{2}=4x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}+8x-2=4x$$

Subtract 4x from both sides.

$$2x^{2}+8x-2-4x=0$$

Combine 8x and -4x to get 4x.

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

Add 2 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 4 by 2.

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

Divide 2 by 2.

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

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.

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

Square 1.

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

Add 1 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.

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

Subtract 1 from both sides of the equation.

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