Solve 5/x-x-7/x+2=x+2/x

$$\frac { 5 } { x } - \frac { x - 7 } { x + 2 } = \frac { x + 2 } { x }$$

Solve for x

$$x=\sqrt{7}+2\approx 4.645751311$$

$$x=2-\sqrt{7}\approx -0.645751311$$

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,x+2.

$$\left(x+2\right)\times 5-x\left(x-7\right)=\left(x+2\right)\left(x+2\right)$$

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

$$\left(x+2\right)\times 5-x\left(x-7\right)=\left(x+2\right)^{2}$$

Use the distributive property to multiply x+2 by 5.

$$5x+10-x\left(x-7\right)=\left(x+2\right)^{2}$$

Use the distributive property to multiply x by x-7.

$$5x+10-\left(x^{2}-7x\right)=\left(x+2\right)^{2}$$

To find the opposite of x^{2}-7x, find the opposite of each term.

$$5x+10-x^{2}+7x=\left(x+2\right)^{2}$$

Combine 5x and 7x to get 12x.

$$12x+10-x^{2}=\left(x+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$12x+10-x^{2}=x^{2}+4x+4$$

Subtract x^{2} from both sides.

$$12x+10-x^{2}-x^{2}=4x+4$$

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

$$12x+10-2x^{2}=4x+4$$

Subtract 4x from both sides.

$$12x+10-2x^{2}-4x=4$$

Combine 12x and -4x to get 8x.

$$8x+10-2x^{2}=4$$

Subtract 4 from both sides.

$$8x+10-2x^{2}-4=0$$

Subtract 4 from 10 to get 6.

$$8x+6-2x^{2}=0$$

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.

$$-2x^{2}+8x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 8 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-8±\sqrt{64+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$x=\frac{-8±\sqrt{64+48}}{2\left(-2\right)}$$

Add 64 to 48.

$$x=\frac{-8±\sqrt{112}}{2\left(-2\right)}$$

Take the square root of 112.

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

Multiply 2 times -2.

$$x=\frac{-8±4\sqrt{7}}{-4}$$

Now solve the equation x=\frac{-8±4\sqrt{7}}{-4} when ± is plus. Add -8 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}-8}{-4}$$

Divide -8+4\sqrt{7} by -4.

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

Now solve the equation x=\frac{-8±4\sqrt{7}}{-4} when ± is minus. Subtract 4\sqrt{7} from -8.

$$x=\frac{-4\sqrt{7}-8}{-4}$$

Divide -8-4\sqrt{7} by -4.

$$x=\sqrt{7}+2$$

The equation is now solved.

$$x=2-\sqrt{7} x=\sqrt{7}+2$$

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,x+2.

$$\left(x+2\right)\times 5-x\left(x-7\right)=\left(x+2\right)\left(x+2\right)$$

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

$$\left(x+2\right)\times 5-x\left(x-7\right)=\left(x+2\right)^{2}$$

Use the distributive property to multiply x+2 by 5.

$$5x+10-x\left(x-7\right)=\left(x+2\right)^{2}$$

Use the distributive property to multiply x by x-7.

$$5x+10-\left(x^{2}-7x\right)=\left(x+2\right)^{2}$$

To find the opposite of x^{2}-7x, find the opposite of each term.

$$5x+10-x^{2}+7x=\left(x+2\right)^{2}$$

Combine 5x and 7x to get 12x.

$$12x+10-x^{2}=\left(x+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$12x+10-x^{2}=x^{2}+4x+4$$

Subtract x^{2} from both sides.

$$12x+10-x^{2}-x^{2}=4x+4$$

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

$$12x+10-2x^{2}=4x+4$$

Subtract 4x from both sides.

$$12x+10-2x^{2}-4x=4$$

Combine 12x and -4x to get 8x.

$$8x+10-2x^{2}=4$$

Subtract 10 from both sides.

$$8x-2x^{2}=4-10$$

Subtract 10 from 4 to get -6.

$$8x-2x^{2}=-6$$

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.

$$-2x^{2}+8x=-6$$

Divide both sides by -2.

$$\frac{-2x^{2}+8x}{-2}=-\frac{6}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{8}{-2}x=-\frac{6}{-2}$$

Divide 8 by -2.

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

Divide -6 by -2.

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

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

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

Square -2.

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

Add 3 to 4.

$$x^{2}-4x+4=7$$

Factor x^{2}-4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

$$x=\sqrt{7}+2 x=2-\sqrt{7}$$