Solve 3600(1/s+75+1/s-75)=14

$$3600 ( \frac { 1 } { s + 75 } + \frac { 1 } { s - 75 } ) = 14$$

Solve for s

$$s = -\frac{75}{7} = -10\frac{5}{7} \approx -10.714285714$$

$$s=525$$

Steps Using the Quadratic Formula

Divide both sides by 3600.

$$\frac{1}{s+75}+\frac{1}{s-75}=\frac{14}{3600}$$

Reduce the fraction \frac{14}{3600} to lowest terms by extracting and canceling out 2.

$$\frac{1}{s+75}+\frac{1}{s-75}=\frac{7}{1800}$$

Variable s cannot be equal to any of the values -75,75 since division by zero is not defined. Multiply both sides of the equation by 1800\left(s-75\right)\left(s+75\right), the least common multiple of s+75,s-75,1800.

$$1800s-135000+1800s+135000=7\left(s-75\right)\left(s+75\right)$$

Combine 1800s and 1800s to get 3600s.

$$3600s-135000+135000=7\left(s-75\right)\left(s+75\right)$$

Add -135000 and 135000 to get 0.

$$3600s=7\left(s-75\right)\left(s+75\right)$$

Use the distributive property to multiply 7 by s-75.

$$3600s=\left(7s-525\right)\left(s+75\right)$$

Use the distributive property to multiply 7s-525 by s+75 and combine like terms.

$$3600s=7s^{2}-39375$$

Subtract 7s^{2} from both sides.

$$3600s-7s^{2}=-39375$$

Add 39375 to both sides.

$$3600s-7s^{2}+39375=0$$

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.

$$-7s^{2}+3600s+39375=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 3600 for b, and 39375 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-3600±\sqrt{3600^{2}-4\left(-7\right)\times 39375}}{2\left(-7\right)}$$

Square 3600.

$$s=\frac{-3600±\sqrt{12960000-4\left(-7\right)\times 39375}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$s=\frac{-3600±\sqrt{12960000+28\times 39375}}{2\left(-7\right)}$$

Multiply 28 times 39375.

$$s=\frac{-3600±\sqrt{12960000+1102500}}{2\left(-7\right)}$$

Add 12960000 to 1102500.

$$s=\frac{-3600±\sqrt{14062500}}{2\left(-7\right)}$$

Take the square root of 14062500.

$$s=\frac{-3600±3750}{2\left(-7\right)}$$

Multiply 2 times -7.

$$s=\frac{-3600±3750}{-14}$$

Now solve the equation s=\frac{-3600±3750}{-14} when ± is plus. Add -3600 to 3750.

$$s=\frac{150}{-14}$$

Reduce the fraction \frac{150}{-14} to lowest terms by extracting and canceling out 2.

$$s=-\frac{75}{7}$$

Now solve the equation s=\frac{-3600±3750}{-14} when ± is minus. Subtract 3750 from -3600.

$$s=-\frac{7350}{-14}$$

Divide -7350 by -14.

$$s=525$$

The equation is now solved.

$$s=-\frac{75}{7} s=525$$

Steps for Completing the Square

Divide both sides by 3600.

$$\frac{1}{s+75}+\frac{1}{s-75}=\frac{14}{3600}$$

Reduce the fraction \frac{14}{3600} to lowest terms by extracting and canceling out 2.

$$\frac{1}{s+75}+\frac{1}{s-75}=\frac{7}{1800}$$

Variable s cannot be equal to any of the values -75,75 since division by zero is not defined. Multiply both sides of the equation by 1800\left(s-75\right)\left(s+75\right), the least common multiple of s+75,s-75,1800.

$$1800s-135000+1800s+135000=7\left(s-75\right)\left(s+75\right)$$

Combine 1800s and 1800s to get 3600s.

$$3600s-135000+135000=7\left(s-75\right)\left(s+75\right)$$

Add -135000 and 135000 to get 0.

$$3600s=7\left(s-75\right)\left(s+75\right)$$

Use the distributive property to multiply 7 by s-75.

$$3600s=\left(7s-525\right)\left(s+75\right)$$

Use the distributive property to multiply 7s-525 by s+75 and combine like terms.

$$3600s=7s^{2}-39375$$

Subtract 7s^{2} from both sides.

$$3600s-7s^{2}=-39375$$

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.

$$-7s^{2}+3600s=-39375$$

Divide both sides by -7.

$$\frac{-7s^{2}+3600s}{-7}=-\frac{39375}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$s^{2}+\frac{3600}{-7}s=-\frac{39375}{-7}$$

Divide 3600 by -7.

$$s^{2}-\frac{3600}{7}s=-\frac{39375}{-7}$$

Divide -39375 by -7.

$$s^{2}-\frac{3600}{7}s=5625$$

Divide -\frac{3600}{7}, the coefficient of the x term, by 2 to get -\frac{1800}{7}. Then add the square of -\frac{1800}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-\frac{3600}{7}s+\left(-\frac{1800}{7}\right)^{2}=5625+\left(-\frac{1800}{7}\right)^{2}$$

Square -\frac{1800}{7} by squaring both the numerator and the denominator of the fraction.

$$s^{2}-\frac{3600}{7}s+\frac{3240000}{49}=5625+\frac{3240000}{49}$$

Add 5625 to \frac{3240000}{49}.

$$s^{2}-\frac{3600}{7}s+\frac{3240000}{49}=\frac{3515625}{49}$$

Factor s^{2}-\frac{3600}{7}s+\frac{3240000}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(s-\frac{1800}{7}\right)^{2}=\frac{3515625}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-\frac{1800}{7}\right)^{2}}=\sqrt{\frac{3515625}{49}}$$

Simplify.

$$s-\frac{1800}{7}=\frac{1875}{7} s-\frac{1800}{7}=-\frac{1875}{7}$$

Add \frac{1800}{7} to both sides of the equation.

$$s=525 s=-\frac{75}{7}$$