Solve 16/x+3-14/x=-2

$$\frac { 16 } { x + 3 } - \frac { 14 } { x } = - 2$$

Solve for x

$$x=-7$$

$$x=3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x+3,x.

$$x\times 16-\left(x+3\right)\times 14=-2x\left(x+3\right)$$

Use the distributive property to multiply x+3 by 14.

$$x\times 16-\left(14x+42\right)=-2x\left(x+3\right)$$

To find the opposite of 14x+42, find the opposite of each term.

$$x\times 16-14x-42=-2x\left(x+3\right)$$

Combine x\times 16 and -14x to get 2x.

$$2x-42=-2x\left(x+3\right)$$

Use the distributive property to multiply -2x by x+3.

$$2x-42=-2x^{2}-6x$$

Add 2x^{2} to both sides.

$$2x-42+2x^{2}=-6x$$

Add 6x to both sides.

$$2x-42+2x^{2}+6x=0$$

Combine 2x and 6x to get 8x.

$$8x-42+2x^{2}=0$$

Divide both sides by 2.

$$4x-21+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+4x-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\left(-21\right)=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 4.

$$a=-3 b=7$$

Rewrite x^{2}+4x-21 as \left(x^{2}-3x\right)+\left(7x-21\right).

$$\left(x^{2}-3x\right)+\left(7x-21\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(x-3\right)+7\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+7\right)$$

To find equation solutions, solve x-3=0 and x+7=0.

$$x=3 x=-7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x+3,x.

$$x\times 16-\left(x+3\right)\times 14=-2x\left(x+3\right)$$

Use the distributive property to multiply x+3 by 14.

$$x\times 16-\left(14x+42\right)=-2x\left(x+3\right)$$

To find the opposite of 14x+42, find the opposite of each term.

$$x\times 16-14x-42=-2x\left(x+3\right)$$

Combine x\times 16 and -14x to get 2x.

$$2x-42=-2x\left(x+3\right)$$

Use the distributive property to multiply -2x by x+3.

$$2x-42=-2x^{2}-6x$$

Add 2x^{2} to both sides.

$$2x-42+2x^{2}=-6x$$

Add 6x to both sides.

$$2x-42+2x^{2}+6x=0$$

Combine 2x and 6x to get 8x.

$$8x-42+2x^{2}=0$$

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.

$$2x^{2}+8x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-42\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-42\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-42\right)}}{2\times 2}$$

Multiply -8 times -42.

$$x=\frac{-8±\sqrt{64+336}}{2\times 2}$$

Add 64 to 336.

$$x=\frac{-8±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-8±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±20}{4}$$

Now solve the equation x=\frac{-8±20}{4} when ± is plus. Add -8 to 20.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{-8±20}{4} when ± is minus. Subtract 20 from -8.

$$x=-\frac{28}{4}$$

Divide -28 by 4.

$$x=-7$$

The equation is now solved.

$$x=3 x=-7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x+3,x.

$$x\times 16-\left(x+3\right)\times 14=-2x\left(x+3\right)$$

Use the distributive property to multiply x+3 by 14.

$$x\times 16-\left(14x+42\right)=-2x\left(x+3\right)$$

To find the opposite of 14x+42, find the opposite of each term.

$$x\times 16-14x-42=-2x\left(x+3\right)$$

Combine x\times 16 and -14x to get 2x.

$$2x-42=-2x\left(x+3\right)$$

Use the distributive property to multiply -2x by x+3.

$$2x-42=-2x^{2}-6x$$

Add 2x^{2} to both sides.

$$2x-42+2x^{2}=-6x$$

Add 6x to both sides.

$$2x-42+2x^{2}+6x=0$$

Combine 2x and 6x to get 8x.

$$8x-42+2x^{2}=0$$

Add 42 to both sides. Anything plus zero gives itself.

$$8x+2x^{2}=42$$

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.

$$2x^{2}+8x=42$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{42}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{42}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{42}{2}$$

Divide 42 by 2.

$$x^{2}+4x=21$$

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.

$$x^{2}+4x+2^{2}=21+2^{2}$$

Square 2.

$$x^{2}+4x+4=21+4$$

Add 21 to 4.

$$x^{2}+4x+4=25$$

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}.

$$\left(x+2\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+2=5 x+2=-5$$

Subtract 2 from both sides of the equation.

$$x=3 x=-7$$