Solve (100+x)(100+x)1/x=204

$$(100+x)(100+x) \frac{ 1 }{ x } =204$$

Solve for x (complex solution)

$$x=2+14\sqrt{51}i\approx 2+99.979998i$$

$$x=-14\sqrt{51}i+2\approx 2-99.979998i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(100+x\right)\left(100+x\right)\times 1=204x$$

Multiply 100+x and 100+x to get \left(100+x\right)^{2}.

$$\left(100+x\right)^{2}\times 1=204x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(100+x\right)^{2}.

$$\left(10000+200x+x^{2}\right)\times 1=204x$$

Use the distributive property to multiply 10000+200x+x^{2} by 1.

$$10000+200x+x^{2}=204x$$

Subtract 204x from both sides.

$$10000+200x+x^{2}-204x=0$$

Combine 200x and -204x to get -4x.

$$10000-4x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-4x+10000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 10000}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 10000}}{2}$$

Multiply -4 times 10000.

$$x=\frac{-\left(-4\right)±\sqrt{16-40000}}{2}$$

Add 16 to -40000.

$$x=\frac{-\left(-4\right)±\sqrt{-39984}}{2}$$

Take the square root of -39984.

$$x=\frac{-\left(-4\right)±28\sqrt{51}i}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±28\sqrt{51}i}{2}$$

Now solve the equation x=\frac{4±28\sqrt{51}i}{2} when ± is plus. Add 4 to 28i\sqrt{51}.

$$x=\frac{4+28\sqrt{51}i}{2}$$

Divide 4+28i\sqrt{51} by 2.

$$x=2+14\sqrt{51}i$$

Now solve the equation x=\frac{4±28\sqrt{51}i}{2} when ± is minus. Subtract 28i\sqrt{51} from 4.

$$x=\frac{-28\sqrt{51}i+4}{2}$$

Divide 4-28i\sqrt{51} by 2.

$$x=-14\sqrt{51}i+2$$

The equation is now solved.

$$x=2+14\sqrt{51}i x=-14\sqrt{51}i+2$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(100+x\right)\left(100+x\right)\times 1=204x$$

Multiply 100+x and 100+x to get \left(100+x\right)^{2}.

$$\left(100+x\right)^{2}\times 1=204x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(100+x\right)^{2}.

$$\left(10000+200x+x^{2}\right)\times 1=204x$$

Use the distributive property to multiply 10000+200x+x^{2} by 1.

$$10000+200x+x^{2}=204x$$

Subtract 204x from both sides.

$$10000+200x+x^{2}-204x=0$$

Combine 200x and -204x to get -4x.

$$10000-4x+x^{2}=0$$

Subtract 10000 from both sides. Anything subtracted from zero gives its negation.

$$-4x+x^{2}=-10000$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-4x=-10000$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-10000+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-10000+4$$

Add -10000 to 4.

$$x^{2}-4x+4=-9996$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=-9996$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-9996}$$

Simplify.

$$x-2=14\sqrt{51}i x-2=-14\sqrt{51}i$$

Add 2 to both sides of the equation.

$$x=2+14\sqrt{51}i x=-14\sqrt{51}i+2$$