Solve frac{6*10^{24}}{r^2}=7.3*10^22/((3.8*10^8)-r)^2

$$\frac { 6 \times 10 ^ { 24 } } { r ^ { 2 } } = \frac { 7.3 \times 10 ^ { 22 } } { ( ( 3.8 \times 10 ^ { 8 } ) - r ) ^ { 2 } }$$

Solve for r

$$r = \frac{7600000000 \sqrt{1095} + 2280000000000}{5927} \approx 427111516.794022202$$

$$r = \frac{2280000000000 - 7600000000 \sqrt{1095}}{5927} \approx 342249036.605674148$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 24 and get 1000000000000000000000000.

$$\frac{6\times 1000000000000000000000000}{r^{2}}=\frac{7.3\times 10^{22}}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Multiply 6 and 1000000000000000000000000 to get 6000000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{7.3\times 10^{22}}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Calculate 10 to the power of 22 and get 10000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{7.3\times 10000000000000000000000}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Multiply 7.3 and 10000000000000000000000 to get 73000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Calculate 10 to the power of 8 and get 100000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(3.8\times 100000000-r\right)^{2}}$$

Multiply 3.8 and 100000000 to get 380000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(380000000-r\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(380000000-r\right)^{2}.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}}$$

Subtract \frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}} from both sides.

$$\frac{6000000000000000000000000}{r^{2}}-\frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}}=0$$

Factor 144400000000000000-760000000r+r^{2}.

$$\frac{6000000000000000000000000}{r^{2}}-\frac{73000000000000000000000}{\left(r-380000000\right)^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r^{2} and \left(r-380000000\right)^{2} is r^{2}\left(r-380000000\right)^{2}. Multiply \frac{6000000000000000000000000}{r^{2}} times \frac{\left(r-380000000\right)^{2}}{\left(r-380000000\right)^{2}}. Multiply \frac{73000000000000000000000}{\left(r-380000000\right)^{2}} times \frac{r^{2}}{r^{2}}.

$$\frac{6000000000000000000000000\left(r-380000000\right)^{2}}{r^{2}\left(r-380000000\right)^{2}}-\frac{73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Since \frac{6000000000000000000000000\left(r-380000000\right)^{2}}{r^{2}\left(r-380000000\right)^{2}} and \frac{73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6000000000000000000000000\left(r-380000000\right)^{2}-73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Do the multiplications in 6000000000000000000000000\left(r-380000000\right)^{2}-73000000000000000000000r^{2}.

$$\frac{6000000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000-73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Combine like terms in 6000000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000-73000000000000000000000r^{2}.

$$\frac{5927000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000}{r^{2}\left(r-380000000\right)^{2}}=0$$

Variable r cannot be equal to any of the values 0,380000000 since division by zero is not defined. Multiply both sides of the equation by r^{2}\left(r-380000000\right)^{2}.

$$5927000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5927000000000000000000000 for a, -4560000000000000000000000000000000 for b, and 866400000000000000000000000000000000000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±\sqrt{\left(-4560000000000000000000000000000000\right)^{2}-4\times 5927000000000000000000000\times 866400000000000000000000000000000000000000}}{2\times 5927000000000000000000000}$$

Square -4560000000000000000000000000000000.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±\sqrt{20793600000000000000000000000000000000000000000000000000000000000000-4\times 5927000000000000000000000\times 866400000000000000000000000000000000000000}}{2\times 5927000000000000000000000}$$

Multiply -4 times 5927000000000000000000000.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±\sqrt{20793600000000000000000000000000000000000000000000000000000000000000-23708000000000000000000000\times 866400000000000000000000000000000000000000}}{2\times 5927000000000000000000000}$$

Multiply -23708000000000000000000000 times 866400000000000000000000000000000000000000.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±\sqrt{20793600000000000000000000000000000000000000000000000000000000000000-20540611200000000000000000000000000000000000000000000000000000000000}}{2\times 5927000000000000000000000}$$

Add 20793600000000000000000000000000000000000000000000000000000000000000 to -20540611200000000000000000000000000000000000000000000000000000000000.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±\sqrt{252988800000000000000000000000000000000000000000000000000000000000}}{2\times 5927000000000000000000000}$$

Take the square root of 252988800000000000000000000000000000000000000000000000000000000000.

$$r=\frac{-\left(-4560000000000000000000000000000000\right)±15200000000000000000000000000000\sqrt{1095}}{2\times 5927000000000000000000000}$$

The opposite of -4560000000000000000000000000000000 is 4560000000000000000000000000000000.

$$r=\frac{4560000000000000000000000000000000±15200000000000000000000000000000\sqrt{1095}}{2\times 5927000000000000000000000}$$

Multiply 2 times 5927000000000000000000000.

$$r=\frac{4560000000000000000000000000000000±15200000000000000000000000000000\sqrt{1095}}{11854000000000000000000000}$$

Now solve the equation r=\frac{4560000000000000000000000000000000±15200000000000000000000000000000\sqrt{1095}}{11854000000000000000000000} when ± is plus. Add 4560000000000000000000000000000000 to 15200000000000000000000000000000\sqrt{1095}.

$$r=\frac{15200000000000000000000000000000\sqrt{1095}+4560000000000000000000000000000000}{11854000000000000000000000}$$

Divide 4560000000000000000000000000000000+15200000000000000000000000000000\sqrt{1095} by 11854000000000000000000000.

$$r=\frac{7600000000\sqrt{1095}[phone]}{5927}$$

Now solve the equation r=\frac{4560000000000000000000000000000000±15200000000000000000000000000000\sqrt{1095}}{11854000000000000000000000} when ± is minus. Subtract 15200000000000000000000000000000\sqrt{1095} from 4560000000000000000000000000000000.

$$r=\frac{4560000000000000000000000000000000-15200000000000000000000000000000\sqrt{1095}}{11854000000000000000000000}$$

Divide 4560000000000000000000000000000000-15200000000000000000000000000000\sqrt{1095} by 11854000000000000000000000.

$$r=\frac{2280000000000-7600000000\sqrt{1095}}{5927}$$

The equation is now solved.

$$r=\frac{7600000000\sqrt{1095}[phone]}{5927} r=\frac{2280000000000-7600000000\sqrt{1095}}{5927}$$

Steps for Completing the Square

Calculate 10 to the power of 24 and get 1000000000000000000000000.

$$\frac{6\times 1000000000000000000000000}{r^{2}}=\frac{7.3\times 10^{22}}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Multiply 6 and 1000000000000000000000000 to get 6000000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{7.3\times 10^{22}}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Calculate 10 to the power of 22 and get 10000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{7.3\times 10000000000000000000000}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Multiply 7.3 and 10000000000000000000000 to get 73000000000000000000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(3.8\times 10^{8}-r\right)^{2}}$$

Calculate 10 to the power of 8 and get 100000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(3.8\times 100000000-r\right)^{2}}$$

Multiply 3.8 and 100000000 to get 380000000.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{\left(380000000-r\right)^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(380000000-r\right)^{2}.

$$\frac{6000000000000000000000000}{r^{2}}=\frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}}$$

Subtract \frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}} from both sides.

$$\frac{6000000000000000000000000}{r^{2}}-\frac{73000000000000000000000}{144400000000000000-760000000r+r^{2}}=0$$

Factor 144400000000000000-760000000r+r^{2}.

$$\frac{6000000000000000000000000}{r^{2}}-\frac{73000000000000000000000}{\left(r-380000000\right)^{2}}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r^{2} and \left(r-380000000\right)^{2} is r^{2}\left(r-380000000\right)^{2}. Multiply \frac{6000000000000000000000000}{r^{2}} times \frac{\left(r-380000000\right)^{2}}{\left(r-380000000\right)^{2}}. Multiply \frac{73000000000000000000000}{\left(r-380000000\right)^{2}} times \frac{r^{2}}{r^{2}}.

$$\frac{6000000000000000000000000\left(r-380000000\right)^{2}}{r^{2}\left(r-380000000\right)^{2}}-\frac{73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Since \frac{6000000000000000000000000\left(r-380000000\right)^{2}}{r^{2}\left(r-380000000\right)^{2}} and \frac{73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6000000000000000000000000\left(r-380000000\right)^{2}-73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Do the multiplications in 6000000000000000000000000\left(r-380000000\right)^{2}-73000000000000000000000r^{2}.

$$\frac{6000000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000-73000000000000000000000r^{2}}{r^{2}\left(r-380000000\right)^{2}}=0$$

Combine like terms in 6000000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000-73000000000000000000000r^{2}.

$$\frac{5927000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000}{r^{2}\left(r-380000000\right)^{2}}=0$$

Variable r cannot be equal to any of the values 0,380000000 since division by zero is not defined. Multiply both sides of the equation by r^{2}\left(r-380000000\right)^{2}.

$$5927000000000000000000000r^{2}-4560000000000000000000000000000000r+866400000000000000000000000000000000000000=0$$

Subtract 866400000000000000000000000000000000000000 from both sides. Anything subtracted from zero gives its negation.

$$5927000000000000000000000r^{2}-4560000000000000000000000000000000r=-866400000000000000000000000000000000000000$$

Divide both sides by 5927000000000000000000000.

$$\frac{5927000000000000000000000r^{2}-4560000000000000000000000000000000r}{5927000000000000000000000}=-\frac{866400000000000000000000000000000000000000}{5927000000000000000000000}$$

Dividing by 5927000000000000000000000 undoes the multiplication by 5927000000000000000000000.

$$r^{2}+\left(-\frac{4560000000000000000000000000000000}{5927000000000000000000000}\right)r=-\frac{866400000000000000000000000000000000000000}{5927000000000000000000000}$$

Reduce the fraction \frac{-4560000000000000000000000000000000}{5927000000000000000000000} to lowest terms by extracting and canceling out 1000000000000000000000.

$$r^{2}-\frac{4560000000000}{5927}r=-\frac{866400000000000000000000000000000000000000}{5927000000000000000000000}$$

Reduce the fraction \frac{-866400000000000000000000000000000000000000}{5927000000000000000000000} to lowest terms by extracting and canceling out 1000000000000000000000.

$$r^{2}-\frac{4560000000000}{5927}r=-\frac{866400000000000000000}{5927}$$

Divide -\frac{4560000000000}{5927}, the coefficient of the x term, by 2 to get -\frac{2280000000000}{5927}. Then add the square of -\frac{2280000000000}{5927} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-\frac{4560000000000}{5927}r+\left(-\frac{2280000000000}{5927}\right)^{2}=-\frac{866400000000000000000}{5927}+\left(-\frac{2280000000000}{5927}\right)^{2}$$

Square -\frac{2280000000000}{5927} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-\frac{4560000000000}{5927}r+\frac{5198400000000000000000000}{35129329}=-\frac{866400000000000000000}{5927}+\frac{5198400000000000000000000}{35129329}$$

Add -\frac{866400000000000000000}{5927} to \frac{5198400000000000000000000}{35129329} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}-\frac{4560000000000}{5927}r+\frac{5198400000000000000000000}{35129329}=\frac{63247200000000000000000}{35129329}$$

Factor r^{2}-\frac{4560000000000}{5927}r+\frac{5198400000000000000000000}{35129329}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-\frac{2280000000000}{5927}\right)^{2}=\frac{63247200000000000000000}{35129329}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{2280000000000}{5927}\right)^{2}}=\sqrt{\frac{63247200000000000000000}{35129329}}$$

Simplify.

$$r-\frac{2280000000000}{5927}=\frac{7600000000\sqrt{1095}}{5927} r-\frac{2280000000000}{5927}=-\frac{7600000000\sqrt{1095}}{5927}$$

Add \frac{2280000000000}{5927} to both sides of the equation.

$$r=\frac{7600000000\sqrt{1095}[phone]}{5927} r=\frac{2280000000000-7600000000\sqrt{1095}}{5927}$$