Solve 0.02=2xleft(1-0.02/1+0.02-100/sqrt{9.96*10^6}right)-1

$$0.02 = 2x \left( \frac{ 1-0.02 }{ 1+0.02 } - \frac{ 100 }{ \sqrt{ 9.96 \times { 10 }^{ 6 } } } \right) -1$$

Solve for x

$$x=\frac{44217\sqrt{249}}{39813250}+\frac{10578267}{19906625}\approx 0.54891943$$

Steps for Solving Linear Equation

Subtract 0.02 from 1 to get 0.98.

$$0.02=2x\left(\frac{0.98}{1+0.02}-\frac{100}{\sqrt{9.96\times 10^{6}}}\right)-1$$

Add 1 and 0.02 to get 1.02.

$$0.02=2x\left(\frac{0.98}{1.02}-\frac{100}{\sqrt{9.96\times 10^{6}}}\right)-1$$

Expand \frac{0.98}{1.02} by multiplying both numerator and the denominator by 100.

$$0.02=2x\left(\frac{98}{102}-\frac{100}{\sqrt{9.96\times 10^{6}}}\right)-1$$

Reduce the fraction \frac{98}{102} to lowest terms by extracting and canceling out 2.

$$0.02=2x\left(\frac{49}{51}-\frac{100}{\sqrt{9.96\times 10^{6}}}\right)-1$$

Calculate 10 to the power of 6 and get 1000000.

$$0.02=2x\left(\frac{49}{51}-\frac{100}{\sqrt{9.96\times 1000000}}\right)-1$$

Multiply 9.96 and 1000000 to get 9960000.

$$0.02=2x\left(\frac{49}{51}-\frac{100}{\sqrt{9960000}}\right)-1$$

Factor 9960000=200^{2}\times 249. Rewrite the square root of the product \sqrt{200^{2}\times 249} as the product of square roots \sqrt{200^{2}}\sqrt{249}. Take the square root of 200^{2}.

$$0.02=2x\left(\frac{49}{51}-\frac{100}{200\sqrt{249}}\right)-1$$

Rationalize the denominator of \frac{100}{200\sqrt{249}} by multiplying numerator and denominator by \sqrt{249}.

$$0.02=2x\left(\frac{49}{51}-\frac{100\sqrt{249}}{200\left(\sqrt{249}\right)^{2}}\right)-1$$

The square of \sqrt{249} is 249.

$$0.02=2x\left(\frac{49}{51}-\frac{100\sqrt{249}}{200\times 249}\right)-1$$

Cancel out 100 in both numerator and denominator.

$$0.02=2x\left(\frac{49}{51}-\frac{\sqrt{249}}{2\times 249}\right)-1$$

Multiply 2 and 249 to get 498.

$$0.02=2x\left(\frac{49}{51}-\frac{\sqrt{249}}{498}\right)-1$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 51 and 498 is 8466. Multiply \frac{49}{51} times \frac{166}{166}. Multiply \frac{\sqrt{249}}{498} times \frac{17}{17}.

$$0.02=2x\left(\frac{49\times 166}{8466}-\frac{17\sqrt{249}}{8466}\right)-1$$

Since \frac{49\times 166}{8466} and \frac{17\sqrt{249}}{8466} have the same denominator, subtract them by subtracting their numerators.

$$0.02=2x\times \frac{49\times 166-17\sqrt{249}}{8466}-1$$

Do the multiplications in 49\times 166-17\sqrt{249}.

$$0.02=2x\times \frac{8134-17\sqrt{249}}{8466}-1$$

Cancel out 8466, the greatest common factor in 2 and 8466.

$$0.02=\frac{8134-17\sqrt{249}}{4233}x-1$$

Divide each term of 8134-17\sqrt{249} by 4233 to get \frac{98}{51}-\frac{1}{249}\sqrt{249}.

$$0.02=\left(\frac{98}{51}-\frac{1}{249}\sqrt{249}\right)x-1$$

Use the distributive property to multiply \frac{98}{51}-\frac{1}{249}\sqrt{249} by x.

$$0.02=\frac{98}{51}x-\frac{1}{249}\sqrt{249}x-1$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{98}{51}x-\frac{1}{249}\sqrt{249}x-1=0.02$$

Add 1 to both sides.

$$\frac{98}{51}x-\frac{1}{249}\sqrt{249}x=0.02+1$$

Add 0.02 and 1 to get 1.02.

$$\frac{98}{51}x-\frac{1}{249}\sqrt{249}x=1.02$$

Combine all terms containing x.

$$\left(\frac{98}{51}-\frac{1}{249}\sqrt{249}\right)x=1.02$$

The equation is in standard form.

$$\left(-\frac{\sqrt{249}}{249}+\frac{98}{51}\right)x=\frac{51}{50}$$

Divide both sides by \frac{98}{51}-\frac{1}{249}\sqrt{249}.

$$\frac{\left(-\frac{\sqrt{249}}{249}+\frac{98}{51}\right)x}{-\frac{\sqrt{249}}{249}+\frac{98}{51}}=\frac{\frac{51}{50}}{-\frac{\sqrt{249}}{249}+\frac{98}{51}}$$

Dividing by \frac{98}{51}-\frac{1}{249}\sqrt{249} undoes the multiplication by \frac{98}{51}-\frac{1}{249}\sqrt{249}.

$$x=\frac{\frac{51}{50}}{-\frac{\sqrt{249}}{249}+\frac{98}{51}}$$

Divide \frac{51}{50} by \frac{98}{51}-\frac{1}{249}\sqrt{249}.

$$x=\frac{44217\sqrt{249}}{39813250}+\frac{10578267}{19906625}$$