Solve 208/x+16+2=216/x

$$\frac { 208 } { x + 16 } + 2 = \frac { 216 } { x }$$

Solve for x

$$x=-48$$

$$x=36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -16,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+16\right), the least common multiple of x+16,x.

$$x\times 208+x\left(x+16\right)\times 2=\left(x+16\right)\times 216$$

Use the distributive property to multiply x by x+16.

$$x\times 208+\left(x^{2}+16x\right)\times 2=\left(x+16\right)\times 216$$

Use the distributive property to multiply x^{2}+16x by 2.

$$x\times 208+2x^{2}+32x=\left(x+16\right)\times 216$$

Combine x\times 208 and 32x to get 240x.

$$240x+2x^{2}=\left(x+16\right)\times 216$$

Use the distributive property to multiply x+16 by 216.

$$240x+2x^{2}=216x+3456$$

Subtract 216x from both sides.

$$240x+2x^{2}-216x=3456$$

Combine 240x and -216x to get 24x.

$$24x+2x^{2}=3456$$

Subtract 3456 from both sides.

$$24x+2x^{2}-3456=0$$

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.

$$2x^{2}+24x-3456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 24 for b, and -3456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 2\left(-3456\right)}}{2\times 2}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 2\left(-3456\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-24±\sqrt{576-8\left(-3456\right)}}{2\times 2}$$

Multiply -8 times -3456.

$$x=\frac{-24±\sqrt{576+27648}}{2\times 2}$$

Add 576 to 27648.

$$x=\frac{-24±\sqrt{28224}}{2\times 2}$$

Take the square root of 28224.

$$x=\frac{-24±168}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-24±168}{4}$$

Now solve the equation x=\frac{-24±168}{4} when ± is plus. Add -24 to 168.

$$x=\frac{144}{4}$$

Divide 144 by 4.

$$x=36$$

Now solve the equation x=\frac{-24±168}{4} when ± is minus. Subtract 168 from -24.

$$x=-\frac{192}{4}$$

Divide -192 by 4.

$$x=-48$$

The equation is now solved.

$$x=36 x=-48$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -16,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+16\right), the least common multiple of x+16,x.

$$x\times 208+x\left(x+16\right)\times 2=\left(x+16\right)\times 216$$

Use the distributive property to multiply x by x+16.

$$x\times 208+\left(x^{2}+16x\right)\times 2=\left(x+16\right)\times 216$$

Use the distributive property to multiply x^{2}+16x by 2.

$$x\times 208+2x^{2}+32x=\left(x+16\right)\times 216$$

Combine x\times 208 and 32x to get 240x.

$$240x+2x^{2}=\left(x+16\right)\times 216$$

Use the distributive property to multiply x+16 by 216.

$$240x+2x^{2}=216x+3456$$

Subtract 216x from both sides.

$$240x+2x^{2}-216x=3456$$

Combine 240x and -216x to get 24x.

$$24x+2x^{2}=3456$$

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.

$$2x^{2}+24x=3456$$

Divide both sides by 2.

$$\frac{2x^{2}+24x}{2}=\frac{3456}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{24}{2}x=\frac{3456}{2}$$

Divide 24 by 2.

$$x^{2}+12x=\frac{3456}{2}$$

Divide 3456 by 2.

$$x^{2}+12x=1728$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=1728+6^{2}$$

Square 6.

$$x^{2}+12x+36=1728+36$$

Add 1728 to 36.

$$x^{2}+12x+36=1764$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=1764$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{1764}$$

Simplify.

$$x+6=42 x+6=-42$$

Subtract 6 from both sides of the equation.

$$x=36 x=-48$$