Solve by Factoring (x-18/x)^2-4(x-18/x)-21=0

${(x - \frac{18}{x})}^{2} - 4 (x - \frac{18}{x}) - 21 = 0$

Step 1

Let $u = x - \frac{18}{x}$ . Substitute $u$ for all occurrences of $x - \frac{18}{x}$ .

$u^{2} - 4 u - 21 = 0$

Step 2

Factor $u^{2} - 4 u - 21$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Write the factored form using these integers.

$(u - 7) (u + 3) = 0$

Step 3

Replace all occurrences of $u$ with $x - \frac{18}{x}$ .

$(x - \frac{18}{x} - 7) (x - \frac{18}{x} + 3) = 0$

Step 4

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

$(\frac{x \cdot x}{x} - \frac{18}{x} - 7) (x - \frac{18}{x} + 3) = 0$

Step 5

Combine the numerators over the common denominator.

$(\frac{x \cdot x - 18}{x} - 7) (x - \frac{18}{x} + 3) = 0$

Step 6

Multiply $x$ by $x$ .

$(\frac{x^{2} - 18}{x} - 7) (x - \frac{18}{x} + 3) = 0$

Step 7

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$(\frac{x^{2} - 18}{x} - 7 \cdot \frac{x}{x}) (x - \frac{18}{x} + 3) = 0$

Step 8

Combine $- 7$ and $\frac{x}{x}$ .

$(\frac{x^{2} - 18}{x} + \frac{- 7 x}{x}) (x - \frac{18}{x} + 3) = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{x^{2} - 18 - 7 x}{x} \cdot (x - \frac{18}{x} + 3) = 0$

Step 10

Factor $x^{2} - 18 - 7 x$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Write the factored form using these integers.

$\frac{(x - 9) (x + 2)}{x} \cdot (x - \frac{18}{x} + 3) = 0$

Step 11

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

$\frac{(x - 9) (x + 2)}{x} \cdot (\frac{x \cdot x}{x} - \frac{18}{x} + 3) = 0$

Step 12

Combine the numerators over the common denominator.

$\frac{(x - 9) (x + 2)}{x} \cdot (\frac{x \cdot x - 18}{x} + 3) = 0$

Step 13

Multiply $x$ by $x$ .

$\frac{(x - 9) (x + 2)}{x} \cdot (\frac{x^{2} - 18}{x} + 3) = 0$

Step 14

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{(x - 9) (x + 2)}{x} \cdot (\frac{x^{2} - 18}{x} + \frac{3 x}{x}) = 0$

Step 15

Combine the numerators over the common denominator.

$\frac{(x - 9) (x + 2)}{x} \cdot \frac{x^{2} - 18 + 3 x}{x} = 0$

Step 16

Factor $x^{2} - 18 + 3 x$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

$\frac{(x - 9) (x + 2)}{x} \cdot \frac{(x - 3) (x + 6)}{x} = 0$

Step 17

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\frac{(x - 9) (x + 2)}{x} = 0$

$\frac{(x - 3) (x + 6)}{x} = 0$

Step 18

Set $\frac{(x - 9) (x + 2)}{x}$ equal to $0$ and solve for $x$ .

Set $\frac{(x - 9) (x + 2)}{x}$ equal to $0$ .

$\frac{(x - 9) (x + 2)}{x} = 0$

Solve $\frac{(x - 9) (x + 2)}{x} = 0$ for $x$ .

Set the numerator equal to zero.

$(x - 9) (x + 2) = 0$

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x + 2 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 9) (x + 2) = 0$ true.

$x = 9, - 2$

Step 19

Set $\frac{(x - 3) (x + 6)}{x}$ equal to $0$ and solve for $x$ .

Set $\frac{(x - 3) (x + 6)}{x}$ equal to $0$ .

$\frac{(x - 3) (x + 6)}{x} = 0$

Solve $\frac{(x - 3) (x + 6)}{x} = 0$ for $x$ .

Set the numerator equal to zero.

$(x - 3) (x + 6) = 0$

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 6 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x - 3) (x + 6) = 0$ true.

$x = 3, - 6$

Step 20

The final solution is all the values that make $\frac{(x - 9) (x + 2)}{x} \cdot \frac{(x - 3) (x + 6)}{x} = 0$ true.

$x = 9, - 2, 3, - 6$

Solve by Factoring (x-18/x)^2-4(x-18/x)-21=0

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