Solve 90/a+5+90/a-5=100

$$\frac { 90 } { a + 5 } + \frac { 90 } { a - 5 } = 100$$

Solve for a

$$a = \frac{\sqrt{2581} + 9}{10} \approx 5.980354318$$

$$a=\frac{9-\sqrt{2581}}{10}\approx -4.180354318$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(a-5\right)\left(a+5\right), the least common multiple of a+5,a-5.

$$\left(a-5\right)\times 90+\left(a+5\right)\times 90=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply a-5 by 90.

$$90a-450+\left(a+5\right)\times 90=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply a+5 by 90.

$$90a-450+90a+450=100\left(a-5\right)\left(a+5\right)$$

Combine 90a and 90a to get 180a.

$$180a-450+450=100\left(a-5\right)\left(a+5\right)$$

Add -450 and 450 to get 0.

$$180a=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply 100 by a-5.

$$180a=\left(100a-500\right)\left(a+5\right)$$

Use the distributive property to multiply 100a-500 by a+5 and combine like terms.

$$180a=100a^{2}-2500$$

Subtract 100a^{2} from both sides.

$$180a-100a^{2}=-2500$$

Add 2500 to both sides.

$$180a-100a^{2}+2500=0$$

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.

$$-100a^{2}+180a+2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 180 for b, and 2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-180±\sqrt{180^{2}-4\left(-100\right)\times 2500}}{2\left(-100\right)}$$

Square 180.

$$a=\frac{-180±\sqrt{32400-4\left(-100\right)\times 2500}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$a=\frac{-180±\sqrt{32400+400\times 2500}}{2\left(-100\right)}$$

Multiply 400 times 2500.

$$a=\frac{-180±\sqrt{32400+1000000}}{2\left(-100\right)}$$

Add 32400 to 1000000.

$$a=\frac{-180±\sqrt{1032400}}{2\left(-100\right)}$$

Take the square root of 1032400.

$$a=\frac{-180±20\sqrt{2581}}{2\left(-100\right)}$$

Multiply 2 times -100.

$$a=\frac{-180±20\sqrt{2581}}{-200}$$

Now solve the equation a=\frac{-180±20\sqrt{2581}}{-200} when ± is plus. Add -180 to 20\sqrt{2581}.

$$a=\frac{20\sqrt{2581}-180}{-200}$$

Divide -180+20\sqrt{2581} by -200.

$$a=\frac{9-\sqrt{2581}}{10}$$

Now solve the equation a=\frac{-180±20\sqrt{2581}}{-200} when ± is minus. Subtract 20\sqrt{2581} from -180.

$$a=\frac{-20\sqrt{2581}-180}{-200}$$

Divide -180-20\sqrt{2581} by -200.

$$a=\frac{\sqrt{2581}+9}{10}$$

The equation is now solved.

$$a=\frac{9-\sqrt{2581}}{10} a=\frac{\sqrt{2581}+9}{10}$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(a-5\right)\left(a+5\right), the least common multiple of a+5,a-5.

$$\left(a-5\right)\times 90+\left(a+5\right)\times 90=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply a-5 by 90.

$$90a-450+\left(a+5\right)\times 90=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply a+5 by 90.

$$90a-450+90a+450=100\left(a-5\right)\left(a+5\right)$$

Combine 90a and 90a to get 180a.

$$180a-450+450=100\left(a-5\right)\left(a+5\right)$$

Add -450 and 450 to get 0.

$$180a=100\left(a-5\right)\left(a+5\right)$$

Use the distributive property to multiply 100 by a-5.

$$180a=\left(100a-500\right)\left(a+5\right)$$

Use the distributive property to multiply 100a-500 by a+5 and combine like terms.

$$180a=100a^{2}-2500$$

Subtract 100a^{2} from both sides.

$$180a-100a^{2}=-2500$$

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.

$$-100a^{2}+180a=-2500$$

Divide both sides by -100.

$$\frac{-100a^{2}+180a}{-100}=-\frac{2500}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$a^{2}+\frac{180}{-100}a=-\frac{2500}{-100}$$

Reduce the fraction \frac{180}{-100} to lowest terms by extracting and canceling out 20.

$$a^{2}-\frac{9}{5}a=-\frac{2500}{-100}$$

Divide -2500 by -100.

$$a^{2}-\frac{9}{5}a=25$$

Divide -\frac{9}{5}, the coefficient of the x term, by 2 to get -\frac{9}{10}. Then add the square of -\frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{9}{5}a+\left(-\frac{9}{10}\right)^{2}=25+\left(-\frac{9}{10}\right)^{2}$$

Square -\frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{9}{5}a+\frac{81}{100}=25+\frac{81}{100}$$

Add 25 to \frac{81}{100}.

$$a^{2}-\frac{9}{5}a+\frac{81}{100}=\frac{2581}{100}$$

Factor a^{2}-\frac{9}{5}a+\frac{81}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{9}{10}\right)^{2}=\frac{2581}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{9}{10}\right)^{2}}=\sqrt{\frac{2581}{100}}$$

Simplify.

$$a-\frac{9}{10}=\frac{\sqrt{2581}}{10} a-\frac{9}{10}=-\frac{\sqrt{2581}}{10}$$

Add \frac{9}{10} to both sides of the equation.

$$a=\frac{\sqrt{2581}+9}{10} a=\frac{9-\sqrt{2581}}{10}$$