Solve (a-320)^2+(frac{sqrt{3}}{3}a)^2=200^2

$$( a - 320 ) ^ { 2 } + ( \frac { \sqrt { 3 } } { 3 } a ) ^ { 2 } = 200 ^ { 2 }$$

Solve for a

$$a=60\sqrt{3}+240\approx 343.923048454$$

$$a=240-60\sqrt{3}\approx 136.076951546$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-320\right)^{2}.

$$a^{2}-640a+102400+\left(\frac{\sqrt{3}}{3}a\right)^{2}=200^{2}$$

Express \frac{\sqrt{3}}{3}a as a single fraction.

$$a^{2}-640a+102400+\left(\frac{\sqrt{3}a}{3}\right)^{2}=200^{2}$$

To raise \frac{\sqrt{3}a}{3} to a power, raise both numerator and denominator to the power and then divide.

$$a^{2}-640a+102400+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2}-640a+102400 times \frac{3^{2}}{3^{2}}.

$$\frac{\left(a^{2}-640a+102400\right)\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{\left(a^{2}-640a+102400\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{2}-640a+102400\right)\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -640a+102400 times \frac{3^{2}}{3^{2}}.

$$a^{2}+\frac{\left(-640a+102400\right)\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{\left(-640a+102400\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$a^{2}+\frac{\left(-640a+102400\right)\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 102400 times \frac{3^{2}}{3^{2}}.

$$a^{2}-640a+\frac{102400\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{102400\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$a^{2}-640a+\frac{102400\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Calculate 200 to the power of 2 and get 40000.

$$a^{2}-640a+\frac{102400\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Calculate 3 to the power of 2 and get 9.

$$a^{2}-640a+\frac{102400\times 9+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Multiply 102400 and 9 to get 921600.

$$a^{2}-640a+\frac{921600+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Expand \left(\sqrt{3}a\right)^{2}.

$$a^{2}-640a+\frac{921600+\left(\sqrt{3}\right)^{2}a^{2}}{3^{2}}=40000$$

The square of \sqrt{3} is 3.

$$a^{2}-640a+\frac{921600+3a^{2}}{3^{2}}=40000$$

Calculate 3 to the power of 2 and get 9.

$$a^{2}-640a+\frac{921600+3a^{2}}{9}=40000$$

Divide each term of 921600+3a^{2} by 9 to get 102400+\frac{1}{3}a^{2}.

$$a^{2}-640a+102400+\frac{1}{3}a^{2}=40000$$

Combine a^{2} and \frac{1}{3}a^{2} to get \frac{4}{3}a^{2}.

$$\frac{4}{3}a^{2}-640a+102400=40000$$

Subtract 40000 from both sides.

$$\frac{4}{3}a^{2}-640a+102400-40000=0$$

Subtract 40000 from 102400 to get 62400.

$$\frac{4}{3}a^{2}-640a+62400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, -640 for b, and 62400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-640\right)±\sqrt{\left(-640\right)^{2}-4\times \frac{4}{3}\times 62400}}{2\times \frac{4}{3}}$$

Square -640.

$$a=\frac{-\left(-640\right)±\sqrt{409600-4\times \frac{4}{3}\times 62400}}{2\times \frac{4}{3}}$$

Multiply -4 times \frac{4}{3}.

$$a=\frac{-\left(-640\right)±\sqrt{409600-\frac{16}{3}\times 62400}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times 62400.

$$a=\frac{-\left(-640\right)±\sqrt{409600-332800}}{2\times \frac{4}{3}}$$

Add 409600 to -332800.

$$a=\frac{-\left(-640\right)±\sqrt{76800}}{2\times \frac{4}{3}}$$

Take the square root of 76800.

$$a=\frac{-\left(-640\right)±160\sqrt{3}}{2\times \frac{4}{3}}$$

The opposite of -640 is 640.

$$a=\frac{640±160\sqrt{3}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$a=\frac{640±160\sqrt{3}}{\frac{8}{3}}$$

Now solve the equation a=\frac{640±160\sqrt{3}}{\frac{8}{3}} when ± is plus. Add 640 to 160\sqrt{3}.

$$a=\frac{160\sqrt{3}+640}{\frac{8}{3}}$$

Divide 640+160\sqrt{3} by \frac{8}{3} by multiplying 640+160\sqrt{3} by the reciprocal of \frac{8}{3}.

$$a=60\sqrt{3}+240$$

Now solve the equation a=\frac{640±160\sqrt{3}}{\frac{8}{3}} when ± is minus. Subtract 160\sqrt{3} from 640.

$$a=\frac{640-160\sqrt{3}}{\frac{8}{3}}$$

Divide 640-160\sqrt{3} by \frac{8}{3} by multiplying 640-160\sqrt{3} by the reciprocal of \frac{8}{3}.

$$a=240-60\sqrt{3}$$

The equation is now solved.

$$a=60\sqrt{3}+240 a=240-60\sqrt{3}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-320\right)^{2}.

$$a^{2}-640a+102400+\left(\frac{\sqrt{3}}{3}a\right)^{2}=200^{2}$$

Express \frac{\sqrt{3}}{3}a as a single fraction.

$$a^{2}-640a+102400+\left(\frac{\sqrt{3}a}{3}\right)^{2}=200^{2}$$

To raise \frac{\sqrt{3}a}{3} to a power, raise both numerator and denominator to the power and then divide.

$$a^{2}-640a+102400+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a^{2}-640a+102400 times \frac{3^{2}}{3^{2}}.

$$\frac{\left(a^{2}-640a+102400\right)\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{\left(a^{2}-640a+102400\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{2}-640a+102400\right)\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -640a+102400 times \frac{3^{2}}{3^{2}}.

$$a^{2}+\frac{\left(-640a+102400\right)\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{\left(-640a+102400\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$a^{2}+\frac{\left(-640a+102400\right)\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 102400 times \frac{3^{2}}{3^{2}}.

$$a^{2}-640a+\frac{102400\times 3^{2}}{3^{2}}+\frac{\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Since \frac{102400\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}a\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

$$a^{2}-640a+\frac{102400\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=200^{2}$$

Calculate 200 to the power of 2 and get 40000.

$$a^{2}-640a+\frac{102400\times 3^{2}+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Calculate 3 to the power of 2 and get 9.

$$a^{2}-640a+\frac{102400\times 9+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Multiply 102400 and 9 to get 921600.

$$a^{2}-640a+\frac{921600+\left(\sqrt{3}a\right)^{2}}{3^{2}}=40000$$

Expand \left(\sqrt{3}a\right)^{2}.

$$a^{2}-640a+\frac{921600+\left(\sqrt{3}\right)^{2}a^{2}}{3^{2}}=40000$$

The square of \sqrt{3} is 3.

$$a^{2}-640a+\frac{921600+3a^{2}}{3^{2}}=40000$$

Calculate 3 to the power of 2 and get 9.

$$a^{2}-640a+\frac{921600+3a^{2}}{9}=40000$$

Divide each term of 921600+3a^{2} by 9 to get 102400+\frac{1}{3}a^{2}.

$$a^{2}-640a+102400+\frac{1}{3}a^{2}=40000$$

Combine a^{2} and \frac{1}{3}a^{2} to get \frac{4}{3}a^{2}.

$$\frac{4}{3}a^{2}-640a+102400=40000$$

Subtract 102400 from both sides.

$$\frac{4}{3}a^{2}-640a=40000-102400$$

Subtract 102400 from 40000 to get -62400.

$$\frac{4}{3}a^{2}-640a=-62400$$

Divide both sides of the equation by \frac{4}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{3}a^{2}-640a}{\frac{4}{3}}=-\frac{62400}{\frac{4}{3}}$$

Dividing by \frac{4}{3} undoes the multiplication by \frac{4}{3}.

$$a^{2}+\left(-\frac{640}{\frac{4}{3}}\right)a=-\frac{62400}{\frac{4}{3}}$$

Divide -640 by \frac{4}{3} by multiplying -640 by the reciprocal of \frac{4}{3}.

$$a^{2}-480a=-\frac{62400}{\frac{4}{3}}$$

Divide -62400 by \frac{4}{3} by multiplying -62400 by the reciprocal of \frac{4}{3}.

$$a^{2}-480a=-46800$$

Divide -480, the coefficient of the x term, by 2 to get -240. Then add the square of -240 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-480a+\left(-240\right)^{2}=-46800+\left(-240\right)^{2}$$

Square -240.

$$a^{2}-480a+57600=-46800+57600$$

Add -46800 to 57600.

$$a^{2}-480a+57600=10800$$

Factor a^{2}-480a+57600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-240\right)^{2}=10800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-240\right)^{2}}=\sqrt{10800}$$

Simplify.

$$a-240=60\sqrt{3} a-240=-60\sqrt{3}$$

Add 240 to both sides of the equation.

$$a=60\sqrt{3}+240 a=240-60\sqrt{3}$$