Solve 640000^+900a^2-left(200+20aright)^2/48000a=16/15

$$\frac{ { 640000 }^{ } +900 { a }^{ 2 } - { \left(200+20a \right) }^{ 2 } }{ 48000a } = \frac{ 16 }{ 15 }$$

Solve for a

$$a = \frac{4 \sqrt{3601} + 296}{5} \approx 107.206666204$$

$$a = \frac{296 - 4 \sqrt{3601}}{5} \approx 11.193333796$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 48000a, the least common multiple of 48000a,15.

$$640000^{1}+900a^{2}-\left(200+20a\right)^{2}=51200a$$

Calculate 640000 to the power of 1 and get 640000.

$$640000+900a^{2}-\left(200+20a\right)^{2}=51200a$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(200+20a\right)^{2}.

$$640000+900a^{2}-\left(40000+8000a+400a^{2}\right)=51200a$$

To find the opposite of 40000+8000a+400a^{2}, find the opposite of each term.

$$640000+900a^{2}-40000-8000a-400a^{2}=51200a$$

Subtract 40000 from 640000 to get 600000.

$$600000+900a^{2}-8000a-400a^{2}=51200a$$

Combine 900a^{2} and -400a^{2} to get 500a^{2}.

$$600000+500a^{2}-8000a=51200a$$

Subtract 51200a from both sides.

$$600000+500a^{2}-8000a-51200a=0$$

Combine -8000a and -51200a to get -59200a.

$$600000+500a^{2}-59200a=0$$

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.

$$500a^{2}-59200a+600000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 500 for a, -59200 for b, and 600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-59200\right)±\sqrt{\left(-59200\right)^{2}-4\times 500\times 600000}}{2\times 500}$$

Square -59200.

$$a=\frac{-\left(-59200\right)±\sqrt{3504640000-4\times 500\times 600000}}{2\times 500}$$

Multiply -4 times 500.

$$a=\frac{-\left(-59200\right)±\sqrt{3504640000-2000\times 600000}}{2\times 500}$$

Multiply -2000 times 600000.

$$a=\frac{-\left(-59200\right)±\sqrt{3504640000-1200000000}}{2\times 500}$$

Add 3504640000 to -1200000000.

$$a=\frac{-\left(-59200\right)±\sqrt{2304640000}}{2\times 500}$$

Take the square root of 2304640000.

$$a=\frac{-\left(-59200\right)±800\sqrt{3601}}{2\times 500}$$

The opposite of -59200 is 59200.

$$a=\frac{59200±800\sqrt{3601}}{2\times 500}$$

Multiply 2 times 500.

$$a=\frac{59200±800\sqrt{3601}}{1000}$$

Now solve the equation a=\frac{59200±800\sqrt{3601}}{1000} when ± is plus. Add 59200 to 800\sqrt{3601}.

$$a=\frac{800\sqrt{3601}+59200}{1000}$$

Divide 59200+800\sqrt{3601} by 1000.

$$a=\frac{4\sqrt{3601}+296}{5}$$

Now solve the equation a=\frac{59200±800\sqrt{3601}}{1000} when ± is minus. Subtract 800\sqrt{3601} from 59200.

$$a=\frac{59200-800\sqrt{3601}}{1000}$$

Divide 59200-800\sqrt{3601} by 1000.

$$a=\frac{296-4\sqrt{3601}}{5}$$

The equation is now solved.

$$a=\frac{4\sqrt{3601}+296}{5} a=\frac{296-4\sqrt{3601}}{5}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 48000a, the least common multiple of 48000a,15.

$$640000^{1}+900a^{2}-\left(200+20a\right)^{2}=51200a$$

Calculate 640000 to the power of 1 and get 640000.

$$640000+900a^{2}-\left(200+20a\right)^{2}=51200a$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(200+20a\right)^{2}.

$$640000+900a^{2}-\left(40000+8000a+400a^{2}\right)=51200a$$

To find the opposite of 40000+8000a+400a^{2}, find the opposite of each term.

$$640000+900a^{2}-40000-8000a-400a^{2}=51200a$$

Subtract 40000 from 640000 to get 600000.

$$600000+900a^{2}-8000a-400a^{2}=51200a$$

Combine 900a^{2} and -400a^{2} to get 500a^{2}.

$$600000+500a^{2}-8000a=51200a$$

Subtract 51200a from both sides.

$$600000+500a^{2}-8000a-51200a=0$$

Combine -8000a and -51200a to get -59200a.

$$600000+500a^{2}-59200a=0$$

Subtract 600000 from both sides. Anything subtracted from zero gives its negation.

$$500a^{2}-59200a=-600000$$

Divide both sides by 500.

$$\frac{500a^{2}-59200a}{500}=-\frac{600000}{500}$$

Dividing by 500 undoes the multiplication by 500.

$$a^{2}+\left(-\frac{59200}{500}\right)a=-\frac{600000}{500}$$

Reduce the fraction \frac{-59200}{500} to lowest terms by extracting and canceling out 100.

$$a^{2}-\frac{592}{5}a=-\frac{600000}{500}$$

Divide -600000 by 500.

$$a^{2}-\frac{592}{5}a=-1200$$

Divide -\frac{592}{5}, the coefficient of the x term, by 2 to get -\frac{296}{5}. Then add the square of -\frac{296}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{592}{5}a+\left(-\frac{296}{5}\right)^{2}=-1200+\left(-\frac{296}{5}\right)^{2}$$

Square -\frac{296}{5} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{592}{5}a+\frac{87616}{25}=-1200+\frac{87616}{25}$$

Add -1200 to \frac{87616}{25}.

$$a^{2}-\frac{592}{5}a+\frac{87616}{25}=\frac{57616}{25}$$

Factor a^{2}-\frac{592}{5}a+\frac{87616}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{296}{5}\right)^{2}=\frac{57616}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{296}{5}\right)^{2}}=\sqrt{\frac{57616}{25}}$$

Simplify.

$$a-\frac{296}{5}=\frac{4\sqrt{3601}}{5} a-\frac{296}{5}=-\frac{4\sqrt{3601}}{5}$$

Add \frac{296}{5} to both sides of the equation.

$$a=\frac{4\sqrt{3601}+296}{5} a=\frac{296-4\sqrt{3601}}{5}$$