Solve 2/x-4-x+1/x+2=1

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

Solve for x

$$x = \frac{\sqrt{177} + 7}{4} \approx 5.076033674$$

$$x=\frac{7-\sqrt{177}}{4}\approx -1.576033674$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x-4,x+2.

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

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

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

Use the distributive property to multiply x-4 by x+1 and combine like terms.

$$2x+4-\left(x^{2}-3x-4\right)=\left(x-4\right)\left(x+2\right)$$

To find the opposite of x^{2}-3x-4, find the opposite of each term.

$$2x+4-x^{2}+3x+4=\left(x-4\right)\left(x+2\right)$$

Combine 2x and 3x to get 5x.

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

Add 4 and 4 to get 8.

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

Use the distributive property to multiply x-4 by x+2 and combine like terms.

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

Subtract x^{2} from both sides.

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

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

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

Add 2x to both sides.

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

Combine 5x and 2x to get 7x.

$$7x+8-2x^{2}=-8$$

Add 8 to both sides.

$$7x+8-2x^{2}+8=0$$

Add 8 and 8 to get 16.

$$7x+16-2x^{2}=0$$

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.

$$-2x^{2}+7x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 7 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\times 16}}{2\left(-2\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\times 16}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-7±\sqrt{49+8\times 16}}{2\left(-2\right)}$$

Multiply 8 times 16.

$$x=\frac{-7±\sqrt{49+128}}{2\left(-2\right)}$$

Add 49 to 128.

$$x=\frac{-7±\sqrt{177}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-7±\sqrt{177}}{-4}$$

Now solve the equation x=\frac{-7±\sqrt{177}}{-4} when ± is plus. Add -7 to \sqrt{177}.

$$x=\frac{\sqrt{177}-7}{-4}$$

Divide -7+\sqrt{177} by -4.

$$x=\frac{7-\sqrt{177}}{4}$$

Now solve the equation x=\frac{-7±\sqrt{177}}{-4} when ± is minus. Subtract \sqrt{177} from -7.

$$x=\frac{-\sqrt{177}-7}{-4}$$

Divide -7-\sqrt{177} by -4.

$$x=\frac{\sqrt{177}+7}{4}$$

The equation is now solved.

$$x=\frac{7-\sqrt{177}}{4} x=\frac{\sqrt{177}+7}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+2\right), the least common multiple of x-4,x+2.

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

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

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

Use the distributive property to multiply x-4 by x+1 and combine like terms.

$$2x+4-\left(x^{2}-3x-4\right)=\left(x-4\right)\left(x+2\right)$$

To find the opposite of x^{2}-3x-4, find the opposite of each term.

$$2x+4-x^{2}+3x+4=\left(x-4\right)\left(x+2\right)$$

Combine 2x and 3x to get 5x.

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

Add 4 and 4 to get 8.

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

Use the distributive property to multiply x-4 by x+2 and combine like terms.

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

Subtract x^{2} from both sides.

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

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

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

Add 2x to both sides.

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

Combine 5x and 2x to get 7x.

$$7x+8-2x^{2}=-8$$

Subtract 8 from both sides.

$$7x-2x^{2}=-8-8$$

Subtract 8 from -8 to get -16.

$$7x-2x^{2}=-16$$

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.

$$-2x^{2}+7x=-16$$

Divide both sides by -2.

$$\frac{-2x^{2}+7x}{-2}=-\frac{16}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{7}{-2}x=-\frac{16}{-2}$$

Divide 7 by -2.

$$x^{2}-\frac{7}{2}x=-\frac{16}{-2}$$

Divide -16 by -2.

$$x^{2}-\frac{7}{2}x=8$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=8+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=8+\frac{49}{16}$$

Add 8 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{177}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{177}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{177}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{\sqrt{177}}{4} x-\frac{7}{4}=-\frac{\sqrt{177}}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{\sqrt{177}+7}{4} x=\frac{7-\sqrt{177}}{4}$$