Solve x^-2+x^-1-30=0

$$x ^ { - 2 } + x ^ { - 1 } - 30 = 0$$

Solve for x

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

$$x=-\frac{1}{6}\approx -0.166666667$$

Steps Using Factoring By Grouping

Reorder the terms.

$$-30+\frac{1}{x}+x^{-2}=0$$

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

$$x\left(-30\right)+1+xx^{-2}=0$$

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

$$x\left(-30\right)+1+x^{-1}=0$$

Reorder the terms.

$$-30x+1+\frac{1}{x}=0$$

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

$$-30xx+x+1=0$$

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

$$-30x^{2}+x+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -30x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=1 ab=-30=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 1.

$$a=6 b=-5$$

Rewrite -30x^{2}+x+1 as \left(-30x^{2}+6x\right)+\left(-5x+1\right).

$$\left(-30x^{2}+6x\right)+\left(-5x+1\right)$$

Factor out -6x in the first and -1 in the second group.

$$-6x\left(5x-1\right)-\left(5x-1\right)$$

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

$$\left(5x-1\right)\left(-6x-1\right)$$

To find equation solutions, solve 5x-1=0 and -6x-1=0.

$$x=\frac{1}{5} x=-\frac{1}{6}$$

Steps Using the Quadratic Formula

Reorder the terms.

$$-30+\frac{1}{x}+x^{-2}=0$$

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

$$x\left(-30\right)+1+xx^{-2}=0$$

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

$$x\left(-30\right)+1+x^{-1}=0$$

Reorder the terms.

$$-30x+1+\frac{1}{x}=0$$

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

$$-30xx+x+1=0$$

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

$$-30x^{2}+x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, 1 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-30\right)}}{2\left(-30\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-30\right)}}{2\left(-30\right)}$$

Multiply -4 times -30.

$$x=\frac{-1±\sqrt{1+120}}{2\left(-30\right)}$$

Add 1 to 120.

$$x=\frac{-1±\sqrt{121}}{2\left(-30\right)}$$

Take the square root of 121.

$$x=\frac{-1±11}{2\left(-30\right)}$$

Multiply 2 times -30.

$$x=\frac{-1±11}{-60}$$

Now solve the equation x=\frac{-1±11}{-60} when ± is plus. Add -1 to 11.

$$x=\frac{10}{-60}$$

Reduce the fraction \frac{10}{-60} to lowest terms by extracting and canceling out 10.

$$x=-\frac{1}{6}$$

Now solve the equation x=\frac{-1±11}{-60} when ± is minus. Subtract 11 from -1.

$$x=-\frac{12}{-60}$$

Reduce the fraction \frac{-12}{-60} to lowest terms by extracting and canceling out 12.

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

The equation is now solved.

$$x=-\frac{1}{6} x=\frac{1}{5}$$

Steps for Completing the Square

Add 30 to both sides. Anything plus zero gives itself.

$$x^{-2}+x^{-1}=30$$

Reorder the terms.

$$\frac{1}{x}+x^{-2}=30$$

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

$$1+xx^{-2}=30x$$

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

$$1+x^{-1}=30x$$

Subtract 30x from both sides.

$$1+x^{-1}-30x=0$$

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

$$x^{-1}-30x=-1$$

Reorder the terms.

$$-30x+\frac{1}{x}=-1$$

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

$$-30xx+1=-x$$

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

$$-30x^{2}+1=-x$$

Add x to both sides.

$$-30x^{2}+1+x=0$$

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

$$-30x^{2}+x=-1$$

Divide both sides by -30.

$$\frac{-30x^{2}+x}{-30}=-\frac{1}{-30}$$

Dividing by -30 undoes the multiplication by -30.

$$x^{2}+\frac{1}{-30}x=-\frac{1}{-30}$$

Divide 1 by -30.

$$x^{2}-\frac{1}{30}x=-\frac{1}{-30}$$

Divide -1 by -30.

$$x^{2}-\frac{1}{30}x=\frac{1}{30}$$

Divide -\frac{1}{30}, the coefficient of the x term, by 2 to get -\frac{1}{60}. Then add the square of -\frac{1}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{30}x+\left(-\frac{1}{60}\right)^{2}=\frac{1}{30}+\left(-\frac{1}{60}\right)^{2}$$

Square -\frac{1}{60} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{30}x+\frac{1}{3600}=\frac{1}{30}+\frac{1}{3600}$$

Add \frac{1}{30} to \frac{1}{3600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{30}x+\frac{1}{3600}=\frac{121}{3600}$$

Factor x^{2}-\frac{1}{30}x+\frac{1}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{60}\right)^{2}=\frac{121}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{60}\right)^{2}}=\sqrt{\frac{121}{3600}}$$

Simplify.

$$x-\frac{1}{60}=\frac{11}{60} x-\frac{1}{60}=-\frac{11}{60}$$

Add \frac{1}{60} to both sides of the equation.

$$x=\frac{1}{5} x=-\frac{1}{6}$$