Algebra Examples

${(x - \frac{40}{x})}^{2} - 3 (x - \frac{40}{x}) - 18 = 0$

Step 1

Let $u = x - \frac{40}{x}$ . Substitute $u$ for all occurrences of $x - \frac{40}{x}$ .

$u^{2} - 3 u - 18 = 0$

Step 2

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

Tap for more steps...

Step 2.1

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 $- 3$ .

$- 6, 3$

Step 2.2

Write the factored form using these integers.

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

Step 3

Replace all occurrences of $u$ with $x - \frac{40}{x}$ .

$(x - \frac{40}{x} - 6) (x - \frac{40}{x} + 3) = 0$

Step 4

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

$(\frac{x \cdot x}{x} - \frac{40}{x} - 6) (x - \frac{40}{x} + 3) = 0$

Step 5

Combine the numerators over the common denominator.

$(\frac{x \cdot x - 40}{x} - 6) (x - \frac{40}{x} + 3) = 0$

Step 6

Multiply $x$ by $x$ .

$(\frac{x^{2} - 40}{x} - 6) (x - \frac{40}{x} + 3) = 0$

Step 7

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

$(\frac{x^{2} - 40}{x} - 6 \cdot \frac{x}{x}) (x - \frac{40}{x} + 3) = 0$

Step 8

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

$(\frac{x^{2} - 40}{x} + \frac{- 6 x}{x}) (x - \frac{40}{x} + 3) = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{x^{2} - 40 - 6 x}{x} \cdot (x - \frac{40}{x} + 3) = 0$

Step 10

Factor $x^{2} - 40 - 6 x$ using the AC method.

Tap for more steps...

Step 10.1

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 $- 40$ and whose sum is $- 6$ .

$- 10, 4$

Step 10.2

Write the factored form using these integers.

$\frac{(x - 10) (x + 4)}{x} \cdot (x - \frac{40}{x} + 3) = 0$

Step 11

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

$\frac{(x - 10) (x + 4)}{x} \cdot (\frac{x \cdot x}{x} - \frac{40}{x} + 3) = 0$

Step 12

Combine the numerators over the common denominator.

$\frac{(x - 10) (x + 4)}{x} \cdot (\frac{x \cdot x - 40}{x} + 3) = 0$

Step 13

Multiply $x$ by $x$ .

$\frac{(x - 10) (x + 4)}{x} \cdot (\frac{x^{2} - 40}{x} + 3) = 0$

Step 14

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

$\frac{(x - 10) (x + 4)}{x} \cdot (\frac{x^{2} - 40}{x} + \frac{3 x}{x}) = 0$

Step 15

Combine the numerators over the common denominator.

$\frac{(x - 10) (x + 4)}{x} \cdot \frac{x^{2} - 40 + 3 x}{x} = 0$

Step 16

Factor $x^{2} - 40 + 3 x$ using the AC method.

Tap for more steps...

Step 16.1

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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 16.2

Write the factored form using these integers.

$\frac{(x - 10) (x + 4)}{x} \cdot \frac{(x - 5) (x + 8)}{x} = 0$

Step 17

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

$\frac{(x - 10) (x + 4)}{x} = 0$

$\frac{(x - 5) (x + 8)}{x} = 0$

Step 18

Set $\frac{(x - 10) (x + 4)}{x}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 18.1

Set $\frac{(x - 10) (x + 4)}{x}$ equal to $0$ .

$\frac{(x - 10) (x + 4)}{x} = 0$

Step 18.2

Solve $\frac{(x - 10) (x + 4)}{x} = 0$ for $x$ .

Tap for more steps...

Step 18.2.1

Set the numerator equal to zero.

$(x - 10) (x + 4) = 0$

Step 18.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 18.2.2.1

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

$x - 10 = 0$

$x + 4 = 0$

Step 18.2.2.2

Set $x - 10$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 18.2.2.2.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 18.2.2.2.2

Add $10$ to both sides of the equation.

$x = 10$

Step 18.2.2.3

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 18.2.2.3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 18.2.2.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 18.2.2.4

The final solution is all the values that make $(x - 10) (x + 4) = 0$ true.

$x = 10, - 4$

Step 19

Set $\frac{(x - 5) (x + 8)}{x}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 19.1

Set $\frac{(x - 5) (x + 8)}{x}$ equal to $0$ .

$\frac{(x - 5) (x + 8)}{x} = 0$

Step 19.2

Solve $\frac{(x - 5) (x + 8)}{x} = 0$ for $x$ .

Tap for more steps...

Step 19.2.1

Set the numerator equal to zero.

$(x - 5) (x + 8) = 0$

Step 19.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 19.2.2.1

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

$x - 5 = 0$

$x + 8 = 0$

Step 19.2.2.2

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 19.2.2.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 19.2.2.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 19.2.2.3

Set $x + 8$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 19.2.2.3.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 19.2.2.3.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 19.2.2.4

The final solution is all the values that make $(x - 5) (x + 8) = 0$ true.

$x = 5, - 8$

Step 20

The final solution is all the values that make $\frac{(x - 10) (x + 4)}{x} \cdot \frac{(x - 5) (x + 8)}{x} = 0$ true.

$x = 10, - 4, 5, - 8$