Solve 1875/(1+x_{j)}+625/(1+x_{j)^2}-3125/(1+x_{j)^3}=0

$$\frac { 1875 } { ( 1 + x _ { j } ) } + \frac { 625 } { ( 1 + x _ { j } ) ^ { 2 } } - \frac { 3125 } { ( 1 + x _ { j } ) ^ { 3 } } = 0$$

Solve for x_j

$$x_{j}=\frac{-\sqrt{61}-7}{6}\approx -2.468374946$$

$$x_{j}=\frac{\sqrt{61}-7}{6}\approx 0.135041613$$

Steps Using the Quadratic Formula

Variable x_{j} cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x_{j}+1\right)^{3}, the least common multiple of 1+x_{j},\left(1+x_{j}\right)^{2},\left(1+x_{j}\right)^{3}.

$$\left(x_{j}+1\right)^{2}\times 1875+\left(x_{j}+1\right)\times 625-3125=0$$

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

$$\left(x_{j}^{2}+2x_{j}+1\right)\times 1875+\left(x_{j}+1\right)\times 625-3125=0$$

Use the distributive property to multiply x_{j}^{2}+2x_{j}+1 by 1875.

$$1875x_{j}^{2}+3750x_{j}+1875+\left(x_{j}+1\right)\times 625-3125=0$$

Use the distributive property to multiply x_{j}+1 by 625.

$$1875x_{j}^{2}+3750x_{j}+1875+625x_{j}+625-3125=0$$

Combine 3750x_{j} and 625x_{j} to get 4375x_{j}.

$$1875x_{j}^{2}+4375x_{j}+1875+625-3125=0$$

Add 1875 and 625 to get 2500.

$$1875x_{j}^{2}+4375x_{j}[phone]=0$$

Subtract 3125 from 2500 to get -625.

$$1875x_{j}^{2}+4375x_{j}-625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1875 for a, 4375 for b, and -625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x_{j}=\frac{-4375±\sqrt{4375^{2}-4\times 1875\left(-625\right)}}{2\times 1875}$$

Square 4375.

$$x_{j}=\frac{-4375±\sqrt{19140625-4\times 1875\left(-625\right)}}{2\times 1875}$$

Multiply -4 times 1875.

$$x_{j}=\frac{-4375±\sqrt{19140625-7500\left(-625\right)}}{2\times 1875}$$

Multiply -7500 times -625.

$$x_{j}=\frac{-4375±\sqrt{19140625+4687500}}{2\times 1875}$$

Add 19140625 to 4687500.

$$x_{j}=\frac{-4375±\sqrt{23828125}}{2\times 1875}$$

Take the square root of 23828125.

$$x_{j}=\frac{-4375±625\sqrt{61}}{2\times 1875}$$

Multiply 2 times 1875.

$$x_{j}=\frac{-4375±625\sqrt{61}}{3750}$$

Now solve the equation x_{j}=\frac{-4375±625\sqrt{61}}{3750} when ± is plus. Add -4375 to 625\sqrt{61}.

$$x_{j}=\frac{625\sqrt{61}-4375}{3750}$$

Divide -4375+625\sqrt{61} by 3750.

$$x_{j}=\frac{\sqrt{61}-7}{6}$$

Now solve the equation x_{j}=\frac{-4375±625\sqrt{61}}{3750} when ± is minus. Subtract 625\sqrt{61} from -4375.

$$x_{j}=\frac{-625\sqrt{61}-4375}{3750}$$

Divide -4375-625\sqrt{61} by 3750.

$$x_{j}=\frac{-\sqrt{61}-7}{6}$$

The equation is now solved.

$$x_{j}=\frac{\sqrt{61}-7}{6} x_{j}=\frac{-\sqrt{61}-7}{6}$$

Steps for Completing the Square

Variable x_{j} cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by \left(x_{j}+1\right)^{3}, the least common multiple of 1+x_{j},\left(1+x_{j}\right)^{2},\left(1+x_{j}\right)^{3}.

$$\left(x_{j}+1\right)^{2}\times 1875+\left(x_{j}+1\right)\times 625-3125=0$$

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

$$\left(x_{j}^{2}+2x_{j}+1\right)\times 1875+\left(x_{j}+1\right)\times 625-3125=0$$

Use the distributive property to multiply x_{j}^{2}+2x_{j}+1 by 1875.

$$1875x_{j}^{2}+3750x_{j}+1875+\left(x_{j}+1\right)\times 625-3125=0$$

Use the distributive property to multiply x_{j}+1 by 625.

$$1875x_{j}^{2}+3750x_{j}+1875+625x_{j}+625-3125=0$$

Combine 3750x_{j} and 625x_{j} to get 4375x_{j}.

$$1875x_{j}^{2}+4375x_{j}+1875+625-3125=0$$

Add 1875 and 625 to get 2500.

$$1875x_{j}^{2}+4375x_{j}[phone]=0$$

Subtract 3125 from 2500 to get -625.

$$1875x_{j}^{2}+4375x_{j}-625=0$$

Add 625 to both sides. Anything plus zero gives itself.

$$1875x_{j}^{2}+4375x_{j}=625$$

Divide both sides by 1875.

$$\frac{1875x_{j}^{2}+4375x_{j}}{1875}=\frac{625}{1875}$$

Dividing by 1875 undoes the multiplication by 1875.

$$x_{j}^{2}+\frac{4375}{1875}x_{j}=\frac{625}{1875}$$

Reduce the fraction \frac{4375}{1875} to lowest terms by extracting and canceling out 625.

$$x_{j}^{2}+\frac{7}{3}x_{j}=\frac{625}{1875}$$

Reduce the fraction \frac{625}{1875} to lowest terms by extracting and canceling out 625.

$$x_{j}^{2}+\frac{7}{3}x_{j}=\frac{1}{3}$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x_{j}^{2}+\frac{7}{3}x_{j}+\left(\frac{7}{6}\right)^{2}=\frac{1}{3}+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x_{j}^{2}+\frac{7}{3}x_{j}+\frac{49}{36}=\frac{1}{3}+\frac{49}{36}$$

Add \frac{1}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x_{j}^{2}+\frac{7}{3}x_{j}+\frac{49}{36}=\frac{61}{36}$$

Factor x_{j}^{2}+\frac{7}{3}x_{j}+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x_{j}+\frac{7}{6}\right)^{2}=\frac{61}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x_{j}+\frac{7}{6}\right)^{2}}=\sqrt{\frac{61}{36}}$$

Simplify.

$$x_{j}+\frac{7}{6}=\frac{\sqrt{61}}{6} x_{j}+\frac{7}{6}=-\frac{\sqrt{61}}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$x_{j}=\frac{\sqrt{61}-7}{6} x_{j}=\frac{-\sqrt{61}-7}{6}$$