Solve 1+x^-1=72x^-2

$$1 + x ^ { - 1 } = 72 x ^ { - 2 }$$

Solve for x

$$x=-9$$

$$x=8$$

Steps Using Factoring

Subtract 72x^{-2} from both sides.

$$1+x^{-1}-72x^{-2}=0$$

Reorder the terms.

$$1+\frac{1}{x}-72x^{-2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$x+1-72x^{-2}x=0$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$x+1-72x^{-1}=0$$

Reorder the terms.

$$x+1-72\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx+x-72=0$$

Multiply x and x to get x^{2}.

$$x^{2}+x-72=0$$

To solve the equation, factor x^{2}+x-72 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=1 ab=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 1.

$$a=-8 b=9$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-8\right)\left(x+9\right)$$

To find equation solutions, solve x-8=0 and x+9=0.

$$x=8 x=-9$$

Steps Using Factoring By Grouping

Subtract 72x^{-2} from both sides.

$$1+x^{-1}-72x^{-2}=0$$

Reorder the terms.

$$1+\frac{1}{x}-72x^{-2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$x+1-72x^{-2}x=0$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$x+1-72x^{-1}=0$$

Reorder the terms.

$$x+1-72\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx+x-72=0$$

Multiply x and x to get x^{2}.

$$x^{2}+x-72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-72\right)=-72$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 1.

$$a=-8 b=9$$

Rewrite x^{2}+x-72 as \left(x^{2}-8x\right)+\left(9x-72\right).

$$\left(x^{2}-8x\right)+\left(9x-72\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-8\right)+9\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+9\right)$$

To find equation solutions, solve x-8=0 and x+9=0.

$$x=8 x=-9$$

Steps Using the Quadratic Formula

Subtract 72x^{-2} from both sides.

$$1+x^{-1}-72x^{-2}=0$$

Reorder the terms.

$$1+\frac{1}{x}-72x^{-2}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$x+1-72x^{-2}x=0$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$x+1-72x^{-1}=0$$

Reorder the terms.

$$x+1-72\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx+x-72=0$$

Multiply x and x to get x^{2}.

$$x^{2}+x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-72\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$x=\frac{-1±\sqrt{1+288}}{2}$$

Add 1 to 288.

$$x=\frac{-1±\sqrt{289}}{2}$$

Take the square root of 289.

$$x=\frac{-1±17}{2}$$

Now solve the equation x=\frac{-1±17}{2} when ± is plus. Add -1 to 17.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-1±17}{2} when ± is minus. Subtract 17 from -1.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=8 x=-9$$

Steps for Completing the Square

Subtract 72x^{-2} from both sides.

$$1+x^{-1}-72x^{-2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x^{-1}-72x^{-2}=-1$$

Reorder the terms.

$$\frac{1}{x}-72x^{-2}=-1$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$1-72x^{-2}x=-x$$

To multiply powers of the same base, add their exponents. Add -2 and 1 to get -1.

$$1-72x^{-1}=-x$$

Add x to both sides.

$$1-72x^{-1}+x=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-72x^{-1}+x=-1$$

Reorder the terms.

$$x-72\times \frac{1}{x}=-1$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx-72=-x$$

Multiply x and x to get x^{2}.

$$x^{2}-72=-x$$

Add x to both sides.

$$x^{2}-72+x=0$$

Add 72 to both sides. Anything plus zero gives itself.

$$x^{2}+x=72$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=72+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=72+\frac{1}{4}$$

Add 72 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{289}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{17}{2} x+\frac{1}{2}=-\frac{17}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=8 x=-9$$