Solve 35/x^2+7x-5/x=x+3/x+7

$$\frac { 35 } { x ^ { 2 } + 7 x } - \frac { 5 } { x } = \frac { x + 3 } { x + 7 }$$

Solve for x

$$x=-8$$

Steps Using Factoring

Variable x cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+7\right), the least common multiple of x^{2}+7x,x,x+7.

$$35-\left(x+7\right)\times 5=x\left(x+3\right)$$

Use the distributive property to multiply x+7 by 5.

$$35-\left(5x+35\right)=x\left(x+3\right)$$

To find the opposite of 5x+35, find the opposite of each term.

$$35-5x-35=x\left(x+3\right)$$

Subtract 35 from 35 to get 0.

$$-5x=x\left(x+3\right)$$

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

$$-5x=x^{2}+3x$$

Subtract x^{2} from both sides.

$$-5x-x^{2}=3x$$

Subtract 3x from both sides.

$$-5x-x^{2}-3x=0$$

Combine -5x and -3x to get -8x.

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

Factor out x.

$$x\left(-8-x\right)=0$$

To find equation solutions, solve x=0 and -8-x=0.

$$x=0 x=-8$$

Variable x cannot be equal to 0.

$$x=-8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+7\right), the least common multiple of x^{2}+7x,x,x+7.

$$35-\left(x+7\right)\times 5=x\left(x+3\right)$$

Use the distributive property to multiply x+7 by 5.

$$35-\left(5x+35\right)=x\left(x+3\right)$$

To find the opposite of 5x+35, find the opposite of each term.

$$35-5x-35=x\left(x+3\right)$$

Subtract 35 from 35 to get 0.

$$-5x=x\left(x+3\right)$$

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

$$-5x=x^{2}+3x$$

Subtract x^{2} from both sides.

$$-5x-x^{2}=3x$$

Subtract 3x from both sides.

$$-5x-x^{2}-3x=0$$

Combine -5x and -3x to get -8x.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of \left(-8\right)^{2}.

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

The opposite of -8 is 8.

$$x=\frac{8±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{8±8}{-2}$$

Now solve the equation x=\frac{8±8}{-2} when ± is plus. Add 8 to 8.

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

Divide 16 by -2.

$$x=-8$$

Now solve the equation x=\frac{8±8}{-2} when ± is minus. Subtract 8 from 8.

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

The equation is now solved.

$$x=-8 x=0$$

Variable x cannot be equal to 0.

$$x=-8$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+7\right), the least common multiple of x^{2}+7x,x,x+7.

$$35-\left(x+7\right)\times 5=x\left(x+3\right)$$

Use the distributive property to multiply x+7 by 5.

$$35-\left(5x+35\right)=x\left(x+3\right)$$

To find the opposite of 5x+35, find the opposite of each term.

$$35-5x-35=x\left(x+3\right)$$

Subtract 35 from 35 to get 0.

$$-5x=x\left(x+3\right)$$

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

$$-5x=x^{2}+3x$$

Subtract x^{2} from both sides.

$$-5x-x^{2}=3x$$

Subtract 3x from both sides.

$$-5x-x^{2}-3x=0$$

Combine -5x and -3x to get -8x.

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

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.

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

Divide both sides by -1.

$$\frac{-x^{2}-8x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=\frac{0}{-1}$$

Divide -8 by -1.

$$x^{2}+8x=\frac{0}{-1}$$

Divide 0 by -1.

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

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 4.

$$x^{2}+8x+16=16$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+4=4 x+4=-4$$

Subtract 4 from both sides of the equation.

$$x=0 x=-8$$

Variable x cannot be equal to 0.

$$x=-8$$