Solve 16div(x-6)-20div(x+6)=1

$$16 \div (x-6)-20 \div (x+6)=1$$

Solve for x

$$x=-18$$

$$x=14$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -6,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x+6\right), the least common multiple of x-6,x+6.

$$\left(x+6\right)\times 16-\left(x-6\right)\times 20=\left(x-6\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by 16.

$$16x+96-\left(x-6\right)\times 20=\left(x-6\right)\left(x+6\right)$$

Use the distributive property to multiply x-6 by 20.

$$16x+96-\left(20x-120\right)=\left(x-6\right)\left(x+6\right)$$

To find the opposite of 20x-120, find the opposite of each term.

$$16x+96-20x+120=\left(x-6\right)\left(x+6\right)$$

Combine 16x and -20x to get -4x.

$$-4x+96+120=\left(x-6\right)\left(x+6\right)$$

Add 96 and 120 to get 216.

$$-4x+216=\left(x-6\right)\left(x+6\right)$$

Consider \left(x-6\right)\left(x+6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 6.

$$-4x+216=x^{2}-36$$

Subtract x^{2} from both sides.

$$-4x+216-x^{2}=-36$$

Add 36 to both sides.

$$-4x+216-x^{2}+36=0$$

Add 216 and 36 to get 252.

$$-4x+252-x^{2}=0$$

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.

$$-x^{2}-4x+252=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 252 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 252}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 252}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 252}}{2\left(-1\right)}$$

Multiply 4 times 252.

$$x=\frac{-\left(-4\right)±\sqrt{16+1008}}{2\left(-1\right)}$$

Add 16 to 1008.

$$x=\frac{-\left(-4\right)±\sqrt{1024}}{2\left(-1\right)}$$

Take the square root of 1024.

$$x=\frac{-\left(-4\right)±32}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±32}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±32}{-2}$$

Now solve the equation x=\frac{4±32}{-2} when ± is plus. Add 4 to 32.

$$x=\frac{36}{-2}$$

Divide 36 by -2.

$$x=-18$$

Now solve the equation x=\frac{4±32}{-2} when ± is minus. Subtract 32 from 4.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=-18 x=14$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -6,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x+6\right), the least common multiple of x-6,x+6.

$$\left(x+6\right)\times 16-\left(x-6\right)\times 20=\left(x-6\right)\left(x+6\right)$$

Use the distributive property to multiply x+6 by 16.

$$16x+96-\left(x-6\right)\times 20=\left(x-6\right)\left(x+6\right)$$

Use the distributive property to multiply x-6 by 20.

$$16x+96-\left(20x-120\right)=\left(x-6\right)\left(x+6\right)$$

To find the opposite of 20x-120, find the opposite of each term.

$$16x+96-20x+120=\left(x-6\right)\left(x+6\right)$$

Combine 16x and -20x to get -4x.

$$-4x+96+120=\left(x-6\right)\left(x+6\right)$$

Add 96 and 120 to get 216.

$$-4x+216=\left(x-6\right)\left(x+6\right)$$

Consider \left(x-6\right)\left(x+6\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 6.

$$-4x+216=x^{2}-36$$

Subtract x^{2} from both sides.

$$-4x+216-x^{2}=-36$$

Subtract 216 from both sides.

$$-4x-x^{2}=-36-216$$

Subtract 216 from -36 to get -252.

$$-4x-x^{2}=-252$$

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.

$$-x^{2}-4x=-252$$

Divide both sides by -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{252}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{252}{-1}$$

Divide -4 by -1.

$$x^{2}+4x=-\frac{252}{-1}$$

Divide -252 by -1.

$$x^{2}+4x=252$$

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.

$$x^{2}+4x+2^{2}=252+2^{2}$$

Square 2.

$$x^{2}+4x+4=252+4$$

Add 252 to 4.

$$x^{2}+4x+4=256$$

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}.

$$\left(x+2\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{256}$$

Simplify.

$$x+2=16 x+2=-16$$

Subtract 2 from both sides of the equation.

$$x=14 x=-18$$