Solve by Factoring 1/(x^2)-6=10/x

$\frac{1}{x^{2}} - 6 = \frac{10}{x}$

Step 1

Subtract $\frac{10}{x}$ from both sides of the equation.

$\frac{1}{x^{2}} - 6 - \frac{10}{x} = 0$

Step 2

Reorder terms.

$- \frac{10}{x} + \frac{1}{x^{2}} - 6 = 0$

Step 3

To write $- \frac{10}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{10}{x} \cdot \frac{x}{x} + \frac{1}{x^{2}} - 6 = 0$

Step 4

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10}{x}$ by $\frac{x}{x}$ .

$- \frac{10 x}{x \cdot x} + \frac{1}{x^{2}} - 6 = 0$

Raise $x$ to the power of $1$ .

$- \frac{10 x}{x \cdot x} + \frac{1}{x^{2}} - 6 = 0$

Raise $x$ to the power of $1$ .

$- \frac{10 x}{x \cdot x} + \frac{1}{x^{2}} - 6 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{10 x}{x^{1 + 1}} + \frac{1}{x^{2}} - 6 = 0$

Add $1$ and $1$ .

$- \frac{10 x}{x^{2}} + \frac{1}{x^{2}} - 6 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 10 x + 1}{x^{2}} - 6 = 0$

Step 6

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- 10 x + 1}{x^{2}} - 6 \cdot \frac{x^{2}}{x^{2}} = 0$

Step 7

Combine $- 6$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- 10 x + 1}{x^{2}} + \frac{- 6 x^{2}}{x^{2}} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- 10 x + 1 - 6 x^{2}}{x^{2}} = 0$

Step 9

Reorder terms.

$\frac{- 6 x^{2} - 10 x + 1}{x^{2}} = 0$

Step 10

Set the numerator equal to zero.

$- 6 x^{2} - 10 x + 1 = 0$

Step 11

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 6$ , $b = - 10$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 6 \cdot 1)}}{2 \cdot - 6}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot 1}}{2 \cdot - 6}$

Multiply $- 4 \cdot - 6 \cdot 1$ .

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot 1}}{2 \cdot - 6}$

Multiply $24$ by $1$ .

$x = \frac{10 \pm \sqrt{100 + 24}}{2 \cdot - 6}$

Add $100$ and $24$ .

$x = \frac{10 \pm \sqrt{124}}{2 \cdot - 6}$

Rewrite $124$ as $2^{2} \cdot 31$ .

Factor $4$ out of $124$ .

$x = \frac{10 \pm \sqrt{4 (31)}}{2 \cdot - 6}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot - 6}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{31}}{2 \cdot - 6}$

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{31}}{- 12}$

Simplify $\frac{10 \pm 2 \sqrt{31}}{- 12}$ .

$x = \frac{5 \pm \sqrt{31}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{31}}{6}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot 1}}{2 \cdot - 6}$

Multiply $- 4 \cdot - 6 \cdot 1$ .

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot 1}}{2 \cdot - 6}$

Multiply $24$ by $1$ .

$x = \frac{10 \pm \sqrt{100 + 24}}{2 \cdot - 6}$

Add $100$ and $24$ .

$x = \frac{10 \pm \sqrt{124}}{2 \cdot - 6}$

Rewrite $124$ as $2^{2} \cdot 31$ .

Factor $4$ out of $124$ .

$x = \frac{10 \pm \sqrt{4 (31)}}{2 \cdot - 6}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot - 6}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{31}}{2 \cdot - 6}$

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{31}}{- 12}$

Simplify $\frac{10 \pm 2 \sqrt{31}}{- 12}$ .

$x = \frac{5 \pm \sqrt{31}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{31}}{6}$

Change the $\pm$ to $+$ .

$x = - \frac{5 + \sqrt{31}}{6}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot 1}}{2 \cdot - 6}$

Multiply $- 4 \cdot - 6 \cdot 1$ .

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot 1}}{2 \cdot - 6}$

Multiply $24$ by $1$ .

$x = \frac{10 \pm \sqrt{100 + 24}}{2 \cdot - 6}$

Add $100$ and $24$ .

$x = \frac{10 \pm \sqrt{124}}{2 \cdot - 6}$

Rewrite $124$ as $2^{2} \cdot 31$ .

Factor $4$ out of $124$ .

$x = \frac{10 \pm \sqrt{4 (31)}}{2 \cdot - 6}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot - 6}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{31}}{2 \cdot - 6}$

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{31}}{- 12}$

Simplify $\frac{10 \pm 2 \sqrt{31}}{- 12}$ .

$x = \frac{5 \pm \sqrt{31}}{- 6}$

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{31}}{6}$

Change the $\pm$ to $-$ .

$x = - \frac{5 - \sqrt{31}}{6}$

The final answer is the combination of both solutions.

$x = - \frac{5 + \sqrt{31}}{6}, - \frac{5 - \sqrt{31}}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 + \sqrt{31}}{6}, - \frac{5 - \sqrt{31}}{6}$

Decimal Form:

$x = - 1.76129406 \dots, 0.09462739 \dots$

Solve by Factoring 1/(x^2)-6=10/x

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