Solve x^-2-18x^-1+32=0

$$x ^ { - 2 } - 18 x ^ { - 1 } + 32 = 0$$

Solve for x

$$x=\frac{1}{16}=0.0625$$

$$x=\frac{1}{2}=0.5$$

Steps Using Factoring By Grouping

Reorder the terms.

$$32-18\times \frac{1}{x}+x^{-2}=0$$

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

$$x\times 32-18+xx^{-2}=0$$

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

$$x\times 32-18+x^{-1}=0$$

Reorder the terms.

$$32x-18+\frac{1}{x}=0$$

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

$$32xx+x\left(-18\right)+1=0$$

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

$$32x^{2}+x\left(-18\right)+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 32x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=-18 ab=32\times 1=32$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 32.

$$-1,-32 -2,-16 -4,-8$$

Calculate the sum for each pair.

$$-1-32=-33 -2-16=-18 -4-8=-12$$

The solution is the pair that gives sum -18.

$$a=-16 b=-2$$

Rewrite 32x^{2}-18x+1 as \left(32x^{2}-16x\right)+\left(-2x+1\right).

$$\left(32x^{2}-16x\right)+\left(-2x+1\right)$$

Factor out 16x in the first and -1 in the second group.

$$16x\left(2x-1\right)-\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(16x-1\right)$$

To find equation solutions, solve 2x-1=0 and 16x-1=0.

$$x=\frac{1}{2} x=\frac{1}{16}$$

Steps Using the Quadratic Formula

Reorder the terms.

$$32-18\times \frac{1}{x}+x^{-2}=0$$

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

$$x\times 32-18+xx^{-2}=0$$

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

$$x\times 32-18+x^{-1}=0$$

Reorder the terms.

$$32x-18+\frac{1}{x}=0$$

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

$$32xx+x\left(-18\right)+1=0$$

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

$$32x^{2}+x\left(-18\right)+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$32x^{2}-18x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 32 for a, -18 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 32}}{2\times 32}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 32}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-18\right)±\sqrt{324-128}}{2\times 32}$$

Add 324 to -128.

$$x=\frac{-\left(-18\right)±\sqrt{196}}{2\times 32}$$

Take the square root of 196.

$$x=\frac{-\left(-18\right)±14}{2\times 32}$$

The opposite of -18 is 18.

$$x=\frac{18±14}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{18±14}{64}$$

Now solve the equation x=\frac{18±14}{64} when ± is plus. Add 18 to 14.

$$x=\frac{32}{64}$$

Reduce the fraction \frac{32}{64} to lowest terms by extracting and canceling out 32.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{18±14}{64} when ± is minus. Subtract 14 from 18.

$$x=\frac{4}{64}$$

Reduce the fraction \frac{4}{64} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{16}$$

The equation is now solved.

$$x=\frac{1}{2} x=\frac{1}{16}$$

Steps for Completing the Square

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$x^{-2}-18x^{-1}=-32$$

Reorder the terms.

$$-18\times \frac{1}{x}+x^{-2}=-32$$

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

$$-18+xx^{-2}=-32x$$

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

$$-18+x^{-1}=-32x$$

Add 32x to both sides.

$$-18+x^{-1}+32x=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$x^{-1}+32x=18$$

Reorder the terms.

$$32x+\frac{1}{x}=18$$

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

$$32xx+1=18x$$

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

$$32x^{2}+1=18x$$

Subtract 18x from both sides.

$$32x^{2}+1-18x=0$$

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

$$32x^{2}-18x=-1$$

Divide both sides by 32.

$$\frac{32x^{2}-18x}{32}=-\frac{1}{32}$$

Dividing by 32 undoes the multiplication by 32.

$$x^{2}+\left(-\frac{18}{32}\right)x=-\frac{1}{32}$$

Reduce the fraction \frac{-18}{32} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{9}{16}x=-\frac{1}{32}$$

Divide -\frac{9}{16}, the coefficient of the x term, by 2 to get -\frac{9}{32}. Then add the square of -\frac{9}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{16}x+\left(-\frac{9}{32}\right)^{2}=-\frac{1}{32}+\left(-\frac{9}{32}\right)^{2}$$

Square -\frac{9}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{16}x+\frac{81}{1024}=-\frac{1}{32}+\frac{81}{1024}$$

Add -\frac{1}{32} to \frac{81}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{16}x+\frac{81}{1024}=\frac{49}{1024}$$

Factor x^{2}-\frac{9}{16}x+\frac{81}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{32}\right)^{2}=\frac{49}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{32}\right)^{2}}=\sqrt{\frac{49}{1024}}$$

Simplify.

$$x-\frac{9}{32}=\frac{7}{32} x-\frac{9}{32}=-\frac{7}{32}$$

Add \frac{9}{32} to both sides of the equation.

$$x=\frac{1}{2} x=\frac{1}{16}$$