Solve x=20*sqrt{1-frac{x^2/910^16}}{2*6*10^-8}

$$x = \frac{ 20 \times \sqrt{ 1- \frac{ { x }^{ 2 } }{ 9 { 10 }^{ 16 } } } }{ 2 \cdot 6 \times { 10 }^{ -8 } }$$

Solve for x

$$x = \frac{750000000 \sqrt{106}}{53} \approx 145692879.353589624$$

Solution Steps

Calculate 10 to the power of 16 and get 10000000000000000.

$$x=\frac{20\sqrt{1-\frac{x^{2}}{9\times 10000000000000000}}}{2\times 6\times 10^{-8}}$$

Multiply 9 and 10000000000000000 to get 90000000000000000.

$$x=\frac{20\sqrt{1-\frac{x^{2}}{90000000000000000}}}{2\times 6\times 10^{-8}}$$

Multiply 2 and 6 to get 12.

$$x=\frac{20\sqrt{1-\frac{x^{2}}{90000000000000000}}}{12\times 10^{-8}}$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$x=\frac{20\sqrt{1-\frac{x^{2}}{90000000000000000}}}{12\times \frac{1}{100000000}}$$

Multiply 12 and \frac{1}{100000000} to get \frac{3}{25000000}.

$$x=\frac{20\sqrt{1-\frac{x^{2}}{90000000000000000}}}{\frac{3}{25000000}}$$

Divide 20\sqrt{1-\frac{x^{2}}{90000000000000000}} by \frac{3}{25000000} to get \frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}.

$$x=\frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}$$

Subtract \frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}} from both sides.

$$x-\frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}=0$$

Subtract x from both sides of the equation.

$$-\frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}=-x$$

Square both sides of the equation.

$$\left(-\frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}\right)^{2}=\left(-x\right)^{2}$$

Expand \left(-\frac{500000000}{3}\sqrt{1-\frac{x^{2}}{90000000000000000}}\right)^{2}.

$$\left(-\frac{500000000}{3}\right)^{2}\left(\sqrt{1-\frac{x^{2}}{90000000000000000}}\right)^{2}=\left(-x\right)^{2}$$

Calculate -\frac{500000000}{3} to the power of 2 and get \frac{250000000000000000}{9}.

$$\frac{250000000000000000}{9}\left(\sqrt{1-\frac{x^{2}}{90000000000000000}}\right)^{2}=\left(-x\right)^{2}$$

Calculate \sqrt{1-\frac{x^{2}}{90000000000000000}} to the power of 2 and get 1-\frac{x^{2}}{90000000000000000}.

$$\frac{250000000000000000}{9}\left(1-\frac{x^{2}}{90000000000000000}\right)=\left(-x\right)^{2}$$

Use the distributive property to multiply \frac{250000000000000000}{9} by 1-\frac{x^{2}}{90000000000000000}.

$$\frac{250000000000000000}{9}-\frac{250000000000000000}{9}\times \frac{x^{2}}{90000000000000000}=\left(-x\right)^{2}$$

Multiply -\frac{250000000000000000}{9} times \frac{x^{2}}{90000000000000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{250000000000000000}{9}+\frac{-250000000000000000x^{2}}{9\times 90000000000000000}=\left(-x\right)^{2}$$

Cancel out 10000000000000000 in both numerator and denominator.

$$\frac{250000000000000000}{9}+\frac{-25x^{2}}{9\times 9}=\left(-x\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 9 and 9\times 9 is 9\times 9. Multiply \frac{250000000000000000}{9} times \frac{9}{9}.

$$\frac{250000000000000000\times 9}{9\times 9}+\frac{-25x^{2}}{9\times 9}=\left(-x\right)^{2}$$

Since \frac{250000000000000000\times 9}{9\times 9} and \frac{-25x^{2}}{9\times 9} have the same denominator, add them by adding their numerators.

$$\frac{250000000000000000\times 9-25x^{2}}{9\times 9}=\left(-x\right)^{2}$$

Do the multiplications in 250000000000000000\times 9-25x^{2}.

$$\frac{2250000000000000000-25x^{2}}{9\times 9}=\left(-x\right)^{2}$$

Multiply 9 and 9 to get 81.

$$\frac{2250000000000000000-25x^{2}}{81}=\left(-x\right)^{2}$$

Expand \left(-x\right)^{2}.

$$\frac{2250000000000000000-25x^{2}}{81}=\left(-1\right)^{2}x^{2}$$

Calculate -1 to the power of 2 and get 1.

$$\frac{2250000000000000000-25x^{2}}{81}=1x^{2}$$

Divide each term of 2250000000000000000-25x^{2} by 81 to get \frac{250000000000000000}{9}-\frac{25}{81}x^{2}.

$$\frac{250000000000000000}{9}-\frac{25}{81}x^{2}=1x^{2}$$

Subtract 1x^{2} from both sides.

$$\frac{250000000000000000}{9}-\frac{25}{81}x^{2}-x^{2}=0$$

Combine -\frac{25}{81}x^{2} and -x^{2} to get -\frac{106}{81}x^{2}.

$$\frac{250000000000000000}{9}-\frac{106}{81}x^{2}=0$$

Subtract \frac{250000000000000000}{9} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{106}{81}x^{2}=-\frac{250000000000000000}{9}$$

Multiply both sides by -\frac{81}{106}, the reciprocal of -\frac{106}{81}.

$$x^{2}=-\frac{250000000000000000}{9}\left(-\frac{81}{106}\right)$$

Multiply -\frac{250000000000000000}{9} and -\frac{81}{106} to get \frac{1125000000000000000}{53}.

$$x^{2}=\frac{1125000000000000000}{53}$$

Take the square root of both sides of the equation.

$$x=\frac{750000000\sqrt{106}}{53} x=-\frac{750000000\sqrt{106}}{53}$$

Substitute \frac{750000000\sqrt{106}}{53} for x in the equation x=\frac{20\sqrt{1-\frac{x^{2}}{9\times 10^{16}}}}{2\times 6\times 10^{-8}}.

$$\frac{750000000\sqrt{106}}{53}=\frac{20\sqrt{1-\frac{\left(\frac{750000000\sqrt{106}}{53}\right)^{2}}{9\times 10^{16}}}}{2\times 6\times 10^{-8}}$$

Simplify. The value x=\frac{750000000\sqrt{106}}{53} satisfies the equation.

$$\frac{750000000}{53}\times 106^{\frac{1}{2}}=\frac{750000000}{53}\times 106^{\frac{1}{2}}$$

Substitute -\frac{750000000\sqrt{106}}{53} for x in the equation x=\frac{20\sqrt{1-\frac{x^{2}}{9\times 10^{16}}}}{2\times 6\times 10^{-8}}.

$$-\frac{750000000\sqrt{106}}{53}=\frac{20\sqrt{1-\frac{\left(-\frac{750000000\sqrt{106}}{53}\right)^{2}}{9\times 10^{16}}}}{2\times 6\times 10^{-8}}$$

Simplify. The value x=-\frac{750000000\sqrt{106}}{53} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{750000000}{53}\times 106^{\frac{1}{2}}=\frac{750000000}{53}\times 106^{\frac{1}{2}}$$

Equation -\frac{500000000\sqrt{-\frac{x^{2}}{90000000000000000}+1}}{3}=-x has a unique solution.

$$x=\frac{750000000\sqrt{106}}{53}$$