Solve 5740.5/5000=left(1+xright)^2

$$\frac{ 5740.5 }{ 5000 } = { \left(1+x \right) }^{ 2 }$$

Solve for x

$$x=\frac{\sqrt{11481}}{100}-1\approx 0.071494284$$

$$x=-\frac{\sqrt{11481}}{100}-1\approx -2.071494284$$

Steps Using the Quadratic Formula

Expand \frac{5740.5}{5000} by multiplying both numerator and the denominator by 10.

$$\frac{57405}{50000}=\left(1+x\right)^{2}$$

Reduce the fraction \frac{57405}{50000} to lowest terms by extracting and canceling out 5.

$$\frac{11481}{10000}=\left(1+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$\frac{11481}{10000}=1+2x+x^{2}$$

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

$$1+2x+x^{2}=\frac{11481}{10000}$$

Subtract \frac{11481}{10000} from both sides.

$$1+2x+x^{2}-\frac{11481}{10000}=0$$

Subtract \frac{11481}{10000} from 1 to get -\frac{1481}{10000}.

$$-\frac{1481}{10000}+2x+x^{2}=0$$

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.

$$x^{2}+2x-\frac{1481}{10000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -\frac{1481}{10000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-\frac{1481}{10000}\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-\frac{1481}{10000}\right)}}{2}$$

Multiply -4 times -\frac{1481}{10000}.

$$x=\frac{-2±\sqrt{4+\frac{1481}{2500}}}{2}$$

Add 4 to \frac{1481}{2500}.

$$x=\frac{-2±\sqrt{\frac{11481}{2500}}}{2}$$

Take the square root of \frac{11481}{2500}.

$$x=\frac{-2±\frac{\sqrt{11481}}{50}}{2}$$

Now solve the equation x=\frac{-2±\frac{\sqrt{11481}}{50}}{2} when ± is plus. Add -2 to \frac{\sqrt{11481}}{50}.

$$x=\frac{\frac{\sqrt{11481}}{50}-2}{2}$$

Divide -2+\frac{\sqrt{11481}}{50} by 2.

$$x=\frac{\sqrt{11481}}{100}-1$$

Now solve the equation x=\frac{-2±\frac{\sqrt{11481}}{50}}{2} when ± is minus. Subtract \frac{\sqrt{11481}}{50} from -2.

$$x=\frac{-\frac{\sqrt{11481}}{50}-2}{2}$$

Divide -2-\frac{\sqrt{11481}}{50} by 2.

$$x=-\frac{\sqrt{11481}}{100}-1$$

The equation is now solved.

$$x=\frac{\sqrt{11481}}{100}-1 x=-\frac{\sqrt{11481}}{100}-1$$

Steps for Completing the Square

Expand \frac{5740.5}{5000} by multiplying both numerator and the denominator by 10.

$$\frac{57405}{50000}=\left(1+x\right)^{2}$$

Reduce the fraction \frac{57405}{50000} to lowest terms by extracting and canceling out 5.

$$\frac{11481}{10000}=\left(1+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$\frac{11481}{10000}=1+2x+x^{2}$$

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

$$1+2x+x^{2}=\frac{11481}{10000}$$

Subtract 1 from both sides.

$$2x+x^{2}=\frac{11481}{10000}-1$$

Subtract 1 from \frac{11481}{10000} to get \frac{1481}{10000}.

$$2x+x^{2}=\frac{1481}{10000}$$

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.

$$x^{2}+2x=\frac{1481}{10000}$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=\frac{1481}{10000}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{1481}{10000}+1$$

Add \frac{1481}{10000} to 1.

$$x^{2}+2x+1=\frac{11481}{10000}$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=\frac{11481}{10000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{11481}{10000}}$$

Simplify.

$$x+1=\frac{\sqrt{11481}}{100} x+1=-\frac{\sqrt{11481}}{100}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{11481}}{100}-1 x=-\frac{\sqrt{11481}}{100}-1$$