Solve 2/0.01=100+{left(0.1w-10^5/wright)}^2

$$\frac{ 2 }{ 0.01 } = 100+ { \left(0.1w- \frac{ { 10 }^{ 5 } }{ w } \right) }^{ 2 }$$

Solve for w

$$w=50-50\sqrt{401}\approx -951.249219725$$

$$w=50\sqrt{401}+50\approx 1051.249219725$$

$$w=50\sqrt{401}-50\approx 951.249219725$$

$$w=-50\sqrt{401}-50\approx -1051.249219725$$

Steps Using the Quadratic Formula

Expand \frac{2}{0.01} by multiplying both numerator and the denominator by 100. Anything divided by one gives itself.

$$200=100+\left(0.1w-\frac{10^{5}}{w}\right)^{2}$$

Calculate 10 to the power of 5 and get 100000.

$$200=100+\left(0.1w-\frac{100000}{w}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(0.1w-\frac{100000}{w}\right)^{2}.

$$200=100+0.01w^{2}+0.2w\left(-\frac{100000}{w}\right)+\left(-\frac{100000}{w}\right)^{2}$$

Express w\left(-\frac{100000}{w}\right) as a single fraction.

$$200=100+0.01w^{2}+0.2\times \frac{-w\times 100000}{w}+\left(-\frac{100000}{w}\right)^{2}$$

Calculate -\frac{100000}{w} to the power of 2 and get \left(\frac{100000}{w}\right)^{2}.

$$200=100+0.01w^{2}+0.2\times \frac{-w\times 100000}{w}+\left(\frac{100000}{w}\right)^{2}$$

Multiply -1 and 100000 to get -100000.

$$200=100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\left(\frac{100000}{w}\right)^{2}$$

To raise \frac{100000}{w} to a power, raise both numerator and denominator to the power and then divide.

$$200=100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\frac{100000^{2}}{w^{2}}$$

Calculate 100000 to the power of 2 and get 10000000000.

$$200=100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\frac{10000000000}{w^{2}}$$

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

$$100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\frac{10000000000}{w^{2}}=200$$

Subtract 200 from both sides.

$$100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\frac{10000000000}{w^{2}}-200=0$$

Subtract 200 from 100 to get -100.

$$-100+0.01w^{2}+0.2\times \frac{-100000w}{w}+\frac{10000000000}{w^{2}}=0$$

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w^{2}, the least common multiple of w,w^{2}.

$$w^{2}\left(-100\right)+0.01w^{2}w^{2}+0.2w\left(-100000\right)w+10000000000=0$$

Reorder the terms.

$$0.01w^{2}w^{2}-100w^{2}-100000\times 0.2ww+10000000000=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$0.01w^{4}-100w^{2}-100000\times 0.2ww+10000000000=0$$

Multiply w and w to get w^{2}.

$$0.01w^{4}-100w^{2}-100000\times 0.2w^{2}[phone]=0$$

Multiply -100000 and 0.2 to get -20000.

$$0.01w^{4}-100w^{2}-20000w^{2}[phone]=0$$

Combine -100w^{2} and -20000w^{2} to get -20100w^{2}.

$$0.01w^{4}-20100w^{2}[phone]=0$$

Substitute t for w^{2}.

$$0.01t^{2}-20100t+10000000000=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 0.01 for a, -20100 for b, and 10000000000 for c in the quadratic formula.

$$t=\frac{-\left(-20100\right)±\sqrt{\left(-20100\right)^{2}-4\times 0.01\times 10000000000}}{0.01\times 2}$$

Do the calculations.

$$t=\frac{20100±100\sqrt{401}}{0.02}$$

Solve the equation t=\frac{20100±100\sqrt{401}}{0.02} when ± is plus and when ± is minus.

$$t=5000\sqrt{401}+1005000 t=1005000-5000\sqrt{401}$$

Since w=t^{2}, the solutions are obtained by evaluating w=±\sqrt{t} for each t.

$$w=50\sqrt{401}+50 w=-\left(50\sqrt{401}+50\right) w=-\left(50-50\sqrt{401}\right) w=50-50\sqrt{401}$$