Solve 1200/a=1200/(a-20)+5

$$\frac { 1200 } { a } = \frac { 1200 } { ( a - 20 ) } + 5$$

Solve for a

$$a=-10\sqrt{47}i+10\approx 10-68.556546004i$$

$$a=10+10\sqrt{47}i\approx 10+68.556546004i$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values 0,20 since division by zero is not defined. Multiply both sides of the equation by a\left(a-20\right), the least common multiple of a,a-20.

$$\left(a-20\right)\times 1200=a\times 1200+a\left(a-20\right)\times 5$$

Use the distributive property to multiply a-20 by 1200.

$$1200a-24000=a\times 1200+a\left(a-20\right)\times 5$$

Use the distributive property to multiply a by a-20.

$$1200a-24000=a\times 1200+\left(a^{2}-20a\right)\times 5$$

Use the distributive property to multiply a^{2}-20a by 5.

$$1200a-24000=a\times 1200+5a^{2}-100a$$

Combine a\times 1200 and -100a to get 1100a.

$$1200a-24000=1100a+5a^{2}$$

Subtract 1100a from both sides.

$$1200a-24000-1100a=5a^{2}$$

Combine 1200a and -1100a to get 100a.

$$100a-24000=5a^{2}$$

Subtract 5a^{2} from both sides.

$$100a-24000-5a^{2}=0$$

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.

$$-5a^{2}+100a-24000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 100 for b, and -24000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-100±\sqrt{100^{2}-4\left(-5\right)\left(-24000\right)}}{2\left(-5\right)}$$

Square 100.

$$a=\frac{-100±\sqrt{10000-4\left(-5\right)\left(-24000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$a=\frac{-100±\sqrt{10000+20\left(-24000\right)}}{2\left(-5\right)}$$

Multiply 20 times -24000.

$$a=\frac{-100±\sqrt{10000-480000}}{2\left(-5\right)}$$

Add 10000 to -480000.

$$a=\frac{-100±\sqrt{-470000}}{2\left(-5\right)}$$

Take the square root of -470000.

$$a=\frac{-100±100\sqrt{47}i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$a=\frac{-100±100\sqrt{47}i}{-10}$$

Now solve the equation a=\frac{-100±100\sqrt{47}i}{-10} when ± is plus. Add -100 to 100i\sqrt{47}.

$$a=\frac{-100+100\sqrt{47}i}{-10}$$

Divide -100+100i\sqrt{47} by -10.

$$a=-10\sqrt{47}i+10$$

Now solve the equation a=\frac{-100±100\sqrt{47}i}{-10} when ± is minus. Subtract 100i\sqrt{47} from -100.

$$a=\frac{-100\sqrt{47}i-100}{-10}$$

Divide -100-100i\sqrt{47} by -10.

$$a=10+10\sqrt{47}i$$

The equation is now solved.

$$a=-10\sqrt{47}i+10 a=10+10\sqrt{47}i$$

Steps for Completing the Square

Variable a cannot be equal to any of the values 0,20 since division by zero is not defined. Multiply both sides of the equation by a\left(a-20\right), the least common multiple of a,a-20.

$$\left(a-20\right)\times 1200=a\times 1200+a\left(a-20\right)\times 5$$

Use the distributive property to multiply a-20 by 1200.

$$1200a-24000=a\times 1200+a\left(a-20\right)\times 5$$

Use the distributive property to multiply a by a-20.

$$1200a-24000=a\times 1200+\left(a^{2}-20a\right)\times 5$$

Use the distributive property to multiply a^{2}-20a by 5.

$$1200a-24000=a\times 1200+5a^{2}-100a$$

Combine a\times 1200 and -100a to get 1100a.

$$1200a-24000=1100a+5a^{2}$$

Subtract 1100a from both sides.

$$1200a-24000-1100a=5a^{2}$$

Combine 1200a and -1100a to get 100a.

$$100a-24000=5a^{2}$$

Subtract 5a^{2} from both sides.

$$100a-24000-5a^{2}=0$$

Add 24000 to both sides. Anything plus zero gives itself.

$$100a-5a^{2}=24000$$

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.

$$-5a^{2}+100a=24000$$

Divide both sides by -5.

$$\frac{-5a^{2}+100a}{-5}=\frac{24000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$a^{2}+\frac{100}{-5}a=\frac{24000}{-5}$$

Divide 100 by -5.

$$a^{2}-20a=\frac{24000}{-5}$$

Divide 24000 by -5.

$$a^{2}-20a=-4800$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-20a+\left(-10\right)^{2}=-4800+\left(-10\right)^{2}$$

Square -10.

$$a^{2}-20a+100=-4800+100$$

Add -4800 to 100.

$$a^{2}-20a+100=-4700$$

Factor a^{2}-20a+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}.

$$\left(a-10\right)^{2}=-4700$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-10\right)^{2}}=\sqrt{-4700}$$

Simplify.

$$a-10=10\sqrt{47}i a-10=-10\sqrt{47}i$$

Add 10 to both sides of the equation.

$$a=10+10\sqrt{47}i a=-10\sqrt{47}i+10$$