Solve 3000=1728/1+x+1728/(1+x)^2

$$3000 = \frac { 1728 } { 1 + x } + \frac { 1728 } { ( 1 + x ) ^ { 2 } }$$

Solve for x

$$x=\frac{-6\sqrt{286}-89}{125}\approx -1.523753657$$

$$x=\frac{6\sqrt{286}-89}{125}\approx 0.099753657$$

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}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$3000\left(x+1\right)^{2}=\left(x+1\right)\times 1728+1728$$

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

$$3000\left(x^{2}+2x+1\right)=\left(x+1\right)\times 1728+1728$$

Use the distributive property to multiply 3000 by x^{2}+2x+1.

$$3000x^{2}+6000x+3000=\left(x+1\right)\times 1728+1728$$

Use the distributive property to multiply x+1 by 1728.

$$3000x^{2}+6000x+3000=1728x+1728+1728$$

Add 1728 and 1728 to get 3456.

$$3000x^{2}+6000x+3000=1728x+3456$$

Subtract 1728x from both sides.

$$3000x^{2}+6000x+3000-1728x=3456$$

Combine 6000x and -1728x to get 4272x.

$$3000x^{2}+4272x+3000=3456$$

Subtract 3456 from both sides.

$$3000x^{2}+4272x+3000-3456=0$$

Subtract 3456 from 3000 to get -456.

$$3000x^{2}+4272x-456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3000 for a, 4272 for b, and -456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4272±\sqrt{4272^{2}-4\times 3000\left(-456\right)}}{2\times 3000}$$

Square 4272.

$$x=\frac{-4272±\sqrt{18249984-4\times 3000\left(-456\right)}}{2\times 3000}$$

Multiply -4 times 3000.

$$x=\frac{-4272±\sqrt{18249984-12000\left(-456\right)}}{2\times 3000}$$

Multiply -12000 times -456.

$$x=\frac{-4272±\sqrt{18249984+5472000}}{2\times 3000}$$

Add 18249984 to 5472000.

$$x=\frac{-4272±\sqrt{23721984}}{2\times 3000}$$

Take the square root of 23721984.

$$x=\frac{-4272±288\sqrt{286}}{2\times 3000}$$

Multiply 2 times 3000.

$$x=\frac{-4272±288\sqrt{286}}{6000}$$

Now solve the equation x=\frac{-4272±288\sqrt{286}}{6000} when ± is plus. Add -4272 to 288\sqrt{286}.

$$x=\frac{288\sqrt{286}-4272}{6000}$$

Divide -4272+288\sqrt{286} by 6000.

$$x=\frac{6\sqrt{286}-89}{125}$$

Now solve the equation x=\frac{-4272±288\sqrt{286}}{6000} when ± is minus. Subtract 288\sqrt{286} from -4272.

$$x=\frac{-288\sqrt{286}-4272}{6000}$$

Divide -4272-288\sqrt{286} by 6000.

$$x=\frac{-6\sqrt{286}-89}{125}$$

The equation is now solved.

$$x=\frac{6\sqrt{286}-89}{125} x=\frac{-6\sqrt{286}-89}{125}$$

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}, the least common multiple of 1+x,\left(1+x\right)^{2}.

$$3000\left(x+1\right)^{2}=\left(x+1\right)\times 1728+1728$$

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

$$3000\left(x^{2}+2x+1\right)=\left(x+1\right)\times 1728+1728$$

Use the distributive property to multiply 3000 by x^{2}+2x+1.

$$3000x^{2}+6000x+3000=\left(x+1\right)\times 1728+1728$$

Use the distributive property to multiply x+1 by 1728.

$$3000x^{2}+6000x+3000=1728x+1728+1728$$

Add 1728 and 1728 to get 3456.

$$3000x^{2}+6000x+3000=1728x+3456$$

Subtract 1728x from both sides.

$$3000x^{2}+6000x+3000-1728x=3456$$

Combine 6000x and -1728x to get 4272x.

$$3000x^{2}+4272x+3000=3456$$

Subtract 3000 from both sides.

$$3000x^{2}+4272x=3456-3000$$

Subtract 3000 from 3456 to get 456.

$$3000x^{2}+4272x=456$$

Divide both sides by 3000.

$$\frac{3000x^{2}+4272x}{3000}=\frac{456}{3000}$$

Dividing by 3000 undoes the multiplication by 3000.

$$x^{2}+\frac{4272}{3000}x=\frac{456}{3000}$$

Reduce the fraction \frac{4272}{3000} to lowest terms by extracting and canceling out 24.

$$x^{2}+\frac{178}{125}x=\frac{456}{3000}$$

Reduce the fraction \frac{456}{3000} to lowest terms by extracting and canceling out 24.

$$x^{2}+\frac{178}{125}x=\frac{19}{125}$$

Divide \frac{178}{125}, the coefficient of the x term, by 2 to get \frac{89}{125}. Then add the square of \frac{89}{125} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{178}{125}x+\left(\frac{89}{125}\right)^{2}=\frac{19}{125}+\left(\frac{89}{125}\right)^{2}$$

Square \frac{89}{125} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{178}{125}x+\frac{7921}{15625}=\frac{19}{125}+\frac{7921}{15625}$$

Add \frac{19}{125} to \frac{7921}{15625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{178}{125}x+\frac{7921}{15625}=\frac{10296}{15625}$$

Factor x^{2}+\frac{178}{125}x+\frac{7921}{15625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{89}{125}\right)^{2}=\frac{10296}{15625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{89}{125}\right)^{2}}=\sqrt{\frac{10296}{15625}}$$

Simplify.

$$x+\frac{89}{125}=\frac{6\sqrt{286}}{125} x+\frac{89}{125}=-\frac{6\sqrt{286}}{125}$$

Subtract \frac{89}{125} from both sides of the equation.

$$x=\frac{6\sqrt{286}-89}{125} x=\frac{-6\sqrt{286}-89}{125}$$