Solve 10/2*820^2*x^2-10/2*800^2*left(x-90right)^2=0

$$\frac{ 10 }{ 2 \times { 820 }^{ 2 } } \times { x }^{ 2 } - \frac{ 10 }{ 2 \times { 800 }^{ 2 } } \times { \left(x-90 \right) }^{ 2 } =0$$

Solve for x

$$x = \frac{410}{9} = 45\frac{5}{9} \approx 45.555555556$$

$$x=3690$$

Steps Using the Quadratic Formula

Calculate 820 to the power of 2 and get 672400.

$$\frac{10}{2\times 672400}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Multiply 2 and 672400 to get 1344800.

$$\frac{10}{1344800}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Reduce the fraction \frac{10}{1344800} to lowest terms by extracting and canceling out 10.

$$\frac{1}{134480}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Calculate 800 to the power of 2 and get 640000.

$$\frac{1}{134480}x^{2}-\frac{10}{2\times 640000}\left(x-90\right)^{2}=0$$

Multiply 2 and 640000 to get 1280000.

$$\frac{1}{134480}x^{2}-\frac{10}{1280000}\left(x-90\right)^{2}=0$$

Reduce the fraction \frac{10}{1280000} to lowest terms by extracting and canceling out 10.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}\left(x-90\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-90\right)^{2}.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}\left(x^{2}-180x+8100\right)=0$$

Use the distributive property to multiply -\frac{1}{128000} by x^{2}-180x+8100.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}x^{2}+\frac{9}{6400}x-\frac{81}{1280}=0$$

Combine \frac{1}{134480}x^{2} and -\frac{1}{128000}x^{2} to get -\frac{81}{215168000}x^{2}.

$$-\frac{81}{215168000}x^{2}+\frac{9}{6400}x-\frac{81}{1280}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{81}{215168000} for a, \frac{9}{6400} for b, and -\frac{81}{1280} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{9}{6400}±\sqrt{\left(\frac{9}{6400}\right)^{2}-4\left(-\frac{81}{215168000}\right)\left(-\frac{81}{1280}\right)}}{2\left(-\frac{81}{215168000}\right)}$$

Square \frac{9}{6400} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{9}{6400}±\sqrt{\frac{81}{40960000}-4\left(-\frac{81}{215168000}\right)\left(-\frac{81}{1280}\right)}}{2\left(-\frac{81}{215168000}\right)}$$

Multiply -4 times -\frac{81}{215168000}.

$$x=\frac{-\frac{9}{6400}±\sqrt{\frac{81}{40960000}+\frac{81}{53792000}\left(-\frac{81}{1280}\right)}}{2\left(-\frac{81}{215168000}\right)}$$

Multiply \frac{81}{53792000} times -\frac{81}{1280} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{6400}±\sqrt{\frac{81}{40960000}-\frac{6561}{68853760000}}}{2\left(-\frac{81}{215168000}\right)}$$

Add \frac{81}{40960000} to -\frac{6561}{68853760000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{9}{6400}±\sqrt{\frac{81}{43033600}}}{2\left(-\frac{81}{215168000}\right)}$$

Take the square root of \frac{81}{43033600}.

$$x=\frac{-\frac{9}{6400}±\frac{9}{6560}}{2\left(-\frac{81}{215168000}\right)}$$

Multiply 2 times -\frac{81}{215168000}.

$$x=\frac{-\frac{9}{6400}±\frac{9}{6560}}{-\frac{81}{107584000}}$$

Now solve the equation x=\frac{-\frac{9}{6400}±\frac{9}{6560}}{-\frac{81}{107584000}} when ± is plus. Add -\frac{9}{6400} to \frac{9}{6560} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{9}{262400}}{-\frac{81}{107584000}}$$

Divide -\frac{9}{262400} by -\frac{81}{107584000} by multiplying -\frac{9}{262400} by the reciprocal of -\frac{81}{107584000}.

$$x=\frac{410}{9}$$

Now solve the equation x=\frac{-\frac{9}{6400}±\frac{9}{6560}}{-\frac{81}{107584000}} when ± is minus. Subtract \frac{9}{6560} from -\frac{9}{6400} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{729}{262400}}{-\frac{81}{107584000}}$$

Divide -\frac{729}{262400} by -\frac{81}{107584000} by multiplying -\frac{729}{262400} by the reciprocal of -\frac{81}{107584000}.

$$x=3690$$

The equation is now solved.

$$x=\frac{410}{9} x=3690$$

Steps for Completing the Square

Calculate 820 to the power of 2 and get 672400.

$$\frac{10}{2\times 672400}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Multiply 2 and 672400 to get 1344800.

$$\frac{10}{1344800}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Reduce the fraction \frac{10}{1344800} to lowest terms by extracting and canceling out 10.

$$\frac{1}{134480}x^{2}-\frac{10}{2\times 800^{2}}\left(x-90\right)^{2}=0$$

Calculate 800 to the power of 2 and get 640000.

$$\frac{1}{134480}x^{2}-\frac{10}{2\times 640000}\left(x-90\right)^{2}=0$$

Multiply 2 and 640000 to get 1280000.

$$\frac{1}{134480}x^{2}-\frac{10}{1280000}\left(x-90\right)^{2}=0$$

Reduce the fraction \frac{10}{1280000} to lowest terms by extracting and canceling out 10.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}\left(x-90\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-90\right)^{2}.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}\left(x^{2}-180x+8100\right)=0$$

Use the distributive property to multiply -\frac{1}{128000} by x^{2}-180x+8100.

$$\frac{1}{134480}x^{2}-\frac{1}{128000}x^{2}+\frac{9}{6400}x-\frac{81}{1280}=0$$

Combine \frac{1}{134480}x^{2} and -\frac{1}{128000}x^{2} to get -\frac{81}{215168000}x^{2}.

$$-\frac{81}{215168000}x^{2}+\frac{9}{6400}x-\frac{81}{1280}=0$$

Add \frac{81}{1280} to both sides. Anything plus zero gives itself.

$$-\frac{81}{215168000}x^{2}+\frac{9}{6400}x=\frac{81}{1280}$$

Divide both sides of the equation by -\frac{81}{215168000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{81}{215168000}x^{2}+\frac{9}{6400}x}{-\frac{81}{215168000}}=\frac{\frac{81}{1280}}{-\frac{81}{215168000}}$$

Dividing by -\frac{81}{215168000} undoes the multiplication by -\frac{81}{215168000}.

$$x^{2}+\frac{\frac{9}{6400}}{-\frac{81}{215168000}}x=\frac{\frac{81}{1280}}{-\frac{81}{215168000}}$$

Divide \frac{9}{6400} by -\frac{81}{215168000} by multiplying \frac{9}{6400} by the reciprocal of -\frac{81}{215168000}.

$$x^{2}-\frac{33620}{9}x=\frac{\frac{81}{1280}}{-\frac{81}{215168000}}$$

Divide \frac{81}{1280} by -\frac{81}{215168000} by multiplying \frac{81}{1280} by the reciprocal of -\frac{81}{215168000}.

$$x^{2}-\frac{33620}{9}x=-168100$$

Divide -\frac{33620}{9}, the coefficient of the x term, by 2 to get -\frac{16810}{9}. Then add the square of -\frac{16810}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{33620}{9}x+\left(-\frac{16810}{9}\right)^{2}=-168100+\left(-\frac{16810}{9}\right)^{2}$$

Square -\frac{16810}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{33620}{9}x+\frac{282576100}{81}=-168100+\frac{282576100}{81}$$

Add -168100 to \frac{282576100}{81}.

$$x^{2}-\frac{33620}{9}x+\frac{282576100}{81}=\frac{268960000}{81}$$

Factor x^{2}-\frac{33620}{9}x+\frac{282576100}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{16810}{9}\right)^{2}=\frac{268960000}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{16810}{9}\right)^{2}}=\sqrt{\frac{268960000}{81}}$$

Simplify.

$$x-\frac{16810}{9}=\frac{16400}{9} x-\frac{16810}{9}=-\frac{16400}{9}$$

Add \frac{16810}{9} to both sides of the equation.

$$x=3690 x=\frac{410}{9}$$