Solve 720+720/(x+1)+720/(x+1)^2=1820

$$720 + \frac { 720 } { ( x + 1 ) } + \frac { 720 } { ( x + 1 ) ^ { 2 } } = 1820$$

Solve for x

$$x=\frac{1}{5}=0.2$$

$$x = -\frac{17}{11} = -1\frac{6}{11} \approx -1.545454545$$

Steps Using Factoring By Grouping

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 x+1,\left(x+1\right)^{2}.

$$\left(x+1\right)^{2}\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

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

$$\left(x^{2}+2x+1\right)\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x^{2}+2x+1 by 720.

$$720x^{2}+1440x+720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x+1 by 720.

$$720x^{2}+1440x+720+720x+720+720=1820\left(x+1\right)^{2}$$

Combine 1440x and 720x to get 2160x.

$$720x^{2}+2160x+720+720+720=1820\left(x+1\right)^{2}$$

Add 720 and 720 to get 1440.

$$720x^{2}+2160x+1440+720=1820\left(x+1\right)^{2}$$

Add 1440 and 720 to get 2160.

$$720x^{2}+2160x+2160=1820\left(x+1\right)^{2}$$

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

$$720x^{2}+2160x+2160=1820\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 1820 by x^{2}+2x+1.

$$720x^{2}+2160x+2160=1820x^{2}+3640x+1820$$

Subtract 1820x^{2} from both sides.

$$720x^{2}+2160x+2160-1820x^{2}=3640x+1820$$

Combine 720x^{2} and -1820x^{2} to get -1100x^{2}.

$$-1100x^{2}+2160x+2160=3640x+1820$$

Subtract 3640x from both sides.

$$-1100x^{2}+2160x+2160-3640x=1820$$

Combine 2160x and -3640x to get -1480x.

$$-1100x^{2}-1480x+2160=1820$$

Subtract 1820 from both sides.

$$-1100x^{2}-1480x+2160-1820=0$$

Subtract 1820 from 2160 to get 340.

$$-1100x^{2}-1480x+340=0$$

Divide both sides by 20.

$$-55x^{2}-74x+17=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -55x^{2}+ax+bx+17. To find a and b, set up a system to be solved.

$$a+b=-74 ab=-55\times 17=-935$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -935.

$$1,-935 5,-187 11,-85 17,-55$$

Calculate the sum for each pair.

$$1-935=-934 5-187=-182 11-85=-74 17-55=-38$$

The solution is the pair that gives sum -74.

$$a=11 b=-85$$

Rewrite -55x^{2}-74x+17 as \left(-55x^{2}+11x\right)+\left(-85x+17\right).

$$\left(-55x^{2}+11x\right)+\left(-85x+17\right)$$

Factor out -11x in the first and -17 in the second group.

$$-11x\left(5x-1\right)-17\left(5x-1\right)$$

Factor out common term 5x-1 by using distributive property.

$$\left(5x-1\right)\left(-11x-17\right)$$

To find equation solutions, solve 5x-1=0 and -11x-17=0.

$$x=\frac{1}{5} x=-\frac{17}{11}$$

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 x+1,\left(x+1\right)^{2}.

$$\left(x+1\right)^{2}\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

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

$$\left(x^{2}+2x+1\right)\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x^{2}+2x+1 by 720.

$$720x^{2}+1440x+720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x+1 by 720.

$$720x^{2}+1440x+720+720x+720+720=1820\left(x+1\right)^{2}$$

Combine 1440x and 720x to get 2160x.

$$720x^{2}+2160x+720+720+720=1820\left(x+1\right)^{2}$$

Add 720 and 720 to get 1440.

$$720x^{2}+2160x+1440+720=1820\left(x+1\right)^{2}$$

Add 1440 and 720 to get 2160.

$$720x^{2}+2160x+2160=1820\left(x+1\right)^{2}$$

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

$$720x^{2}+2160x+2160=1820\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 1820 by x^{2}+2x+1.

$$720x^{2}+2160x+2160=1820x^{2}+3640x+1820$$

Subtract 1820x^{2} from both sides.

$$720x^{2}+2160x+2160-1820x^{2}=3640x+1820$$

Combine 720x^{2} and -1820x^{2} to get -1100x^{2}.

$$-1100x^{2}+2160x+2160=3640x+1820$$

Subtract 3640x from both sides.

$$-1100x^{2}+2160x+2160-3640x=1820$$

Combine 2160x and -3640x to get -1480x.

$$-1100x^{2}-1480x+2160=1820$$

Subtract 1820 from both sides.

$$-1100x^{2}-1480x+2160-1820=0$$

Subtract 1820 from 2160 to get 340.

$$-1100x^{2}-1480x+340=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1100 for a, -1480 for b, and 340 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1480\right)±\sqrt{\left(-1480\right)^{2}-4\left(-1100\right)\times 340}}{2\left(-1100\right)}$$

Square -1480.

$$x=\frac{-\left(-1480\right)±\sqrt{2190400-4\left(-1100\right)\times 340}}{2\left(-1100\right)}$$

Multiply -4 times -1100.

$$x=\frac{-\left(-1480\right)±\sqrt{2190400+4400\times 340}}{2\left(-1100\right)}$$

Multiply 4400 times 340.

$$x=\frac{-\left(-1480\right)±\sqrt{2190400+1496000}}{2\left(-1100\right)}$$

Add 2190400 to 1496000.

$$x=\frac{-\left(-1480\right)±\sqrt{3686400}}{2\left(-1100\right)}$$

Take the square root of 3686400.

$$x=\frac{-\left(-1480\right)±1920}{2\left(-1100\right)}$$

The opposite of -1480 is 1480.

$$x=\frac{1480±1920}{2\left(-1100\right)}$$

Multiply 2 times -1100.

$$x=\frac{1480±1920}{-2200}$$

Now solve the equation x=\frac{1480±1920}{-2200} when ± is plus. Add 1480 to 1920.

$$x=\frac{3400}{-2200}$$

Reduce the fraction \frac{3400}{-2200} to lowest terms by extracting and canceling out 200.

$$x=-\frac{17}{11}$$

Now solve the equation x=\frac{1480±1920}{-2200} when ± is minus. Subtract 1920 from 1480.

$$x=-\frac{440}{-2200}$$

Reduce the fraction \frac{-440}{-2200} to lowest terms by extracting and canceling out 440.

$$x=\frac{1}{5}$$

The equation is now solved.

$$x=-\frac{17}{11} x=\frac{1}{5}$$

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 x+1,\left(x+1\right)^{2}.

$$\left(x+1\right)^{2}\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

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

$$\left(x^{2}+2x+1\right)\times 720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x^{2}+2x+1 by 720.

$$720x^{2}+1440x+720+\left(x+1\right)\times 720+720=1820\left(x+1\right)^{2}$$

Use the distributive property to multiply x+1 by 720.

$$720x^{2}+1440x+720+720x+720+720=1820\left(x+1\right)^{2}$$

Combine 1440x and 720x to get 2160x.

$$720x^{2}+2160x+720+720+720=1820\left(x+1\right)^{2}$$

Add 720 and 720 to get 1440.

$$720x^{2}+2160x+1440+720=1820\left(x+1\right)^{2}$$

Add 1440 and 720 to get 2160.

$$720x^{2}+2160x+2160=1820\left(x+1\right)^{2}$$

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

$$720x^{2}+2160x+2160=1820\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 1820 by x^{2}+2x+1.

$$720x^{2}+2160x+2160=1820x^{2}+3640x+1820$$

Subtract 1820x^{2} from both sides.

$$720x^{2}+2160x+2160-1820x^{2}=3640x+1820$$

Combine 720x^{2} and -1820x^{2} to get -1100x^{2}.

$$-1100x^{2}+2160x+2160=3640x+1820$$

Subtract 3640x from both sides.

$$-1100x^{2}+2160x+2160-3640x=1820$$

Combine 2160x and -3640x to get -1480x.

$$-1100x^{2}-1480x+2160=1820$$

Subtract 2160 from both sides.

$$-1100x^{2}-1480x=1820-2160$$

Subtract 2160 from 1820 to get -340.

$$-1100x^{2}-1480x=-340$$

Divide both sides by -1100.

$$\frac{-1100x^{2}-1480x}{-1100}=-\frac{340}{-1100}$$

Dividing by -1100 undoes the multiplication by -1100.

$$x^{2}+\left(-\frac{1480}{-1100}\right)x=-\frac{340}{-1100}$$

Reduce the fraction \frac{-1480}{-1100} to lowest terms by extracting and canceling out 20.

$$x^{2}+\frac{74}{55}x=-\frac{340}{-1100}$$

Reduce the fraction \frac{-340}{-1100} to lowest terms by extracting and canceling out 20.

$$x^{2}+\frac{74}{55}x=\frac{17}{55}$$

Divide \frac{74}{55}, the coefficient of the x term, by 2 to get \frac{37}{55}. Then add the square of \frac{37}{55} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{74}{55}x+\left(\frac{37}{55}\right)^{2}=\frac{17}{55}+\left(\frac{37}{55}\right)^{2}$$

Square \frac{37}{55} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{74}{55}x+\frac{1369}{3025}=\frac{17}{55}+\frac{1369}{3025}$$

Add \frac{17}{55} to \frac{1369}{3025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{74}{55}x+\frac{1369}{3025}=\frac{2304}{3025}$$

Factor x^{2}+\frac{74}{55}x+\frac{1369}{3025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{37}{55}\right)^{2}=\frac{2304}{3025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{37}{55}\right)^{2}}=\sqrt{\frac{2304}{3025}}$$

Simplify.

$$x+\frac{37}{55}=\frac{48}{55} x+\frac{37}{55}=-\frac{48}{55}$$

Subtract \frac{37}{55} from both sides of the equation.

$$x=\frac{1}{5} x=-\frac{17}{11}$$