Solve 3920/x+2+224=3920/x

$$\frac{ 3920 }{ x+2 } +224 = \frac{ 3920 }{ x }$$

Solve for x

$$x=-7$$

$$x=5$$

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+2,x.

$$x\times 3920+x\left(x+2\right)\times 224=\left(x+2\right)\times 3920$$

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

$$x\times 3920+\left(x^{2}+2x\right)\times 224=\left(x+2\right)\times 3920$$

Use the distributive property to multiply x^{2}+2x by 224.

$$x\times 3920+224x^{2}+448x=\left(x+2\right)\times 3920$$

Combine x\times 3920 and 448x to get 4368x.

$$4368x+224x^{2}=\left(x+2\right)\times 3920$$

Use the distributive property to multiply x+2 by 3920.

$$4368x+224x^{2}=3920x+7840$$

Subtract 3920x from both sides.

$$4368x+224x^{2}-3920x=7840$$

Combine 4368x and -3920x to get 448x.

$$448x+224x^{2}=7840$$

Subtract 7840 from both sides.

$$448x+224x^{2}-7840=0$$

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.

$$224x^{2}+448x-7840=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 224 for a, 448 for b, and -7840 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-448±\sqrt{448^{2}-4\times 224\left(-7840\right)}}{2\times 224}$$

Square 448.

$$x=\frac{-448±\sqrt{200704-4\times 224\left(-7840\right)}}{2\times 224}$$

Multiply -4 times 224.

$$x=\frac{-448±\sqrt{200704-896\left(-7840\right)}}{2\times 224}$$

Multiply -896 times -7840.

$$x=\frac{-448±\sqrt{200704+7024640}}{2\times 224}$$

Add 200704 to 7024640.

$$x=\frac{-448±\sqrt{7225344}}{2\times 224}$$

Take the square root of 7225344.

$$x=\frac{-448±2688}{2\times 224}$$

Multiply 2 times 224.

$$x=\frac{-448±2688}{448}$$

Now solve the equation x=\frac{-448±2688}{448} when ± is plus. Add -448 to 2688.

$$x=\frac{2240}{448}$$

Divide 2240 by 448.

$$x=5$$

Now solve the equation x=\frac{-448±2688}{448} when ± is minus. Subtract 2688 from -448.

$$x=-\frac{3136}{448}$$

Divide -3136 by 448.

$$x=-7$$

The equation is now solved.

$$x=5 x=-7$$

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+2,x.

$$x\times 3920+x\left(x+2\right)\times 224=\left(x+2\right)\times 3920$$

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

$$x\times 3920+\left(x^{2}+2x\right)\times 224=\left(x+2\right)\times 3920$$

Use the distributive property to multiply x^{2}+2x by 224.

$$x\times 3920+224x^{2}+448x=\left(x+2\right)\times 3920$$

Combine x\times 3920 and 448x to get 4368x.

$$4368x+224x^{2}=\left(x+2\right)\times 3920$$

Use the distributive property to multiply x+2 by 3920.

$$4368x+224x^{2}=3920x+7840$$

Subtract 3920x from both sides.

$$4368x+224x^{2}-3920x=7840$$

Combine 4368x and -3920x to get 448x.

$$448x+224x^{2}=7840$$

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.

$$224x^{2}+448x=7840$$

Divide both sides by 224.

$$\frac{224x^{2}+448x}{224}=\frac{7840}{224}$$

Dividing by 224 undoes the multiplication by 224.

$$x^{2}+\frac{448}{224}x=\frac{7840}{224}$$

Divide 448 by 224.

$$x^{2}+2x=\frac{7840}{224}$$

Divide 7840 by 224.

$$x^{2}+2x=35$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=35+1^{2}$$

Square 1.

$$x^{2}+2x+1=35+1$$

Add 35 to 1.

$$x^{2}+2x+1=36$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+1=6 x+1=-6$$

Subtract 1 from both sides of the equation.

$$x=5 x=-7$$