Solve 126/x+2+98/x-2=14

$$\frac { 126 } { x + 2 } + \frac { 98 } { x - 2 } = 14$$

Solve for x

$$x=16$$

$$x=0$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)\times 126+\left(x+2\right)\times 98=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x-2 by 126.

$$126x-252+\left(x+2\right)\times 98=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 98.

$$126x-252+98x+196=14\left(x-2\right)\left(x+2\right)$$

Combine 126x and 98x to get 224x.

$$224x-252+196=14\left(x-2\right)\left(x+2\right)$$

Add -252 and 196 to get -56.

$$224x-56=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 14 by x-2.

$$224x-56=\left(14x-28\right)\left(x+2\right)$$

Use the distributive property to multiply 14x-28 by x+2 and combine like terms.

$$224x-56=14x^{2}-56$$

Subtract 14x^{2} from both sides.

$$224x-56-14x^{2}=-56$$

Add 56 to both sides.

$$224x-56-14x^{2}+56=0$$

Add -56 and 56 to get 0.

$$224x-14x^{2}=0$$

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.

$$-14x^{2}+224x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -14 for a, 224 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-224±\sqrt{224^{2}}}{2\left(-14\right)}$$

Take the square root of 224^{2}.

$$x=\frac{-224±224}{2\left(-14\right)}$$

Multiply 2 times -14.

$$x=\frac{-224±224}{-28}$$

Now solve the equation x=\frac{-224±224}{-28} when ± is plus. Add -224 to 224.

$$x=\frac{0}{-28}$$

Divide 0 by -28.

$$x=0$$

Now solve the equation x=\frac{-224±224}{-28} when ± is minus. Subtract 224 from -224.

$$x=-\frac{448}{-28}$$

Divide -448 by -28.

$$x=16$$

The equation is now solved.

$$x=0 x=16$$

Steps for Completing the Square

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

$$\left(x-2\right)\times 126+\left(x+2\right)\times 98=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x-2 by 126.

$$126x-252+\left(x+2\right)\times 98=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 98.

$$126x-252+98x+196=14\left(x-2\right)\left(x+2\right)$$

Combine 126x and 98x to get 224x.

$$224x-252+196=14\left(x-2\right)\left(x+2\right)$$

Add -252 and 196 to get -56.

$$224x-56=14\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 14 by x-2.

$$224x-56=\left(14x-28\right)\left(x+2\right)$$

Use the distributive property to multiply 14x-28 by x+2 and combine like terms.

$$224x-56=14x^{2}-56$$

Subtract 14x^{2} from both sides.

$$224x-56-14x^{2}=-56$$

Add 56 to both sides.

$$224x-14x^{2}=-56+56$$

Add -56 and 56 to get 0.

$$224x-14x^{2}=0$$

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.

$$-14x^{2}+224x=0$$

Divide both sides by -14.

$$\frac{-14x^{2}+224x}{-14}=\frac{0}{-14}$$

Dividing by -14 undoes the multiplication by -14.

$$x^{2}+\frac{224}{-14}x=\frac{0}{-14}$$

Divide 224 by -14.

$$x^{2}-16x=\frac{0}{-14}$$

Divide 0 by -14.

$$x^{2}-16x=0$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=64$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-8=8 x-8=-8$$

Add 8 to both sides of the equation.

$$x=16 x=0$$