Solve 1500=2000/(1+x)^2

$$1500 = \frac { 2000 } { ( 1 + x ) ^ { 2 } }$$

Solve for x

$$x=-\frac{2\sqrt{3}}{3}-1\approx -2.154700538$$

$$x=\frac{2\sqrt{3}}{3}-1\approx 0.154700538$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}.

$$1500\left(x+1\right)^{2}=2000$$

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

$$1500\left(x^{2}+2x+1\right)=2000$$

Use the distributive property to multiply 1500 by x^{2}+2x+1.

$$1500x^{2}+3000x+1500=2000$$

Subtract 2000 from both sides.

$$1500x^{2}+3000x+1500-2000=0$$

Subtract 2000 from 1500 to get -500.

$$1500x^{2}+3000x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1500 for a, 3000 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3000±\sqrt{3000^{2}-4\times 1500\left(-500\right)}}{2\times 1500}$$

Square 3000.

$$x=\frac{-3000±\sqrt{9000000-4\times 1500\left(-500\right)}}{2\times 1500}$$

Multiply -4 times 1500.

$$x=\frac{-3000±\sqrt{9000000-6000\left(-500\right)}}{2\times 1500}$$

Multiply -6000 times -500.

$$x=\frac{-3000±\sqrt{9000000+3000000}}{2\times 1500}$$

Add 9000000 to 3000000.

$$x=\frac{-3000±\sqrt{12000000}}{2\times 1500}$$

Take the square root of 12000000.

$$x=\frac{-3000±2000\sqrt{3}}{2\times 1500}$$

Multiply 2 times 1500.

$$x=\frac{-3000±2000\sqrt{3}}{3000}$$

Now solve the equation x=\frac{-3000±2000\sqrt{3}}{3000} when ± is plus. Add -3000 to 2000\sqrt{3}.

$$x=\frac{2000\sqrt{3}-3000}{3000}$$

Divide -3000+2000\sqrt{3} by 3000.

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

Now solve the equation x=\frac{-3000±2000\sqrt{3}}{3000} when ± is minus. Subtract 2000\sqrt{3} from -3000.

$$x=\frac{-2000\sqrt{3}-3000}{3000}$$

Divide -3000-2000\sqrt{3} by 3000.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)^{2}.

$$1500\left(x+1\right)^{2}=2000$$

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

$$1500\left(x^{2}+2x+1\right)=2000$$

Use the distributive property to multiply 1500 by x^{2}+2x+1.

$$1500x^{2}+3000x+1500=2000$$

Subtract 1500 from both sides.

$$1500x^{2}+3000x=2000-1500$$

Subtract 1500 from 2000 to get 500.

$$1500x^{2}+3000x=500$$

Divide both sides by 1500.

$$\frac{1500x^{2}+3000x}{1500}=\frac{500}{1500}$$

Dividing by 1500 undoes the multiplication by 1500.

$$x^{2}+\frac{3000}{1500}x=\frac{500}{1500}$$

Divide 3000 by 1500.

$$x^{2}+2x=\frac{500}{1500}$$

Reduce the fraction \frac{500}{1500} to lowest terms by extracting and canceling out 500.

$$x^{2}+2x=\frac{1}{3}$$

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.

$$x^{2}+2x+1^{2}=\frac{1}{3}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{1}{3}+1$$

Add \frac{1}{3} to 1.

$$x^{2}+2x+1=\frac{4}{3}$$

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}.

$$\left(x+1\right)^{2}=\frac{4}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{4}{3}}$$

Simplify.

$$x+1=\frac{2\sqrt{3}}{3} x+1=-\frac{2\sqrt{3}}{3}$$

Subtract 1 from both sides of the equation.

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