Algebra Examples

$\frac{90}{x} - \frac{90}{x - 5} = - \frac{1}{4}$

Step 1

Add $\frac{1}{4}$ to both sides of the equation.

$\frac{90}{x} - \frac{90}{x - 5} + \frac{1}{4} = 0$

Step 2

Simplify $\frac{90}{x} - \frac{90}{x - 5} + \frac{1}{4}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{90}{x}$ by $\frac{(x - 5) \cdot 4}{(x - 5) \cdot 4}$ .

$\frac{90}{x} \cdot \frac{(x - 5) \cdot 4}{(x - 5) \cdot 4} - \frac{90}{x - 5} + \frac{1}{4} = 0$

Step 2.1.2

Multiply $\frac{90}{x}$ by $\frac{(x - 5) \cdot 4}{(x - 5) \cdot 4}$ .

$\frac{90 ((x - 5) \cdot 4)}{x ((x - 5) \cdot 4)} - \frac{90}{x - 5} + \frac{1}{4} = 0$

Step 2.1.3

Multiply $\frac{90}{x - 5}$ by $\frac{x \cdot 4}{x \cdot 4}$ .

$\frac{90 ((x - 5) \cdot 4)}{x ((x - 5) \cdot 4)} - (\frac{90}{x - 5} \cdot \frac{x \cdot 4}{x \cdot 4}) + \frac{1}{4} = 0$

Step 2.1.4

Multiply $\frac{90}{x - 5}$ by $\frac{x \cdot 4}{x \cdot 4}$ .

$\frac{90 ((x - 5) \cdot 4)}{x ((x - 5) \cdot 4)} - \frac{90 (x \cdot 4)}{(x - 5) (x \cdot 4)} + \frac{1}{4} = 0$

Step 2.1.5

Multiply $\frac{1}{4}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{90 ((x - 5) \cdot 4)}{x ((x - 5) \cdot 4)} - \frac{90 (x \cdot 4)}{(x - 5) (x \cdot 4)} + \frac{1}{4} \cdot \frac{x (x - 5)}{x (x - 5)} = 0$

Step 2.1.6

Multiply $\frac{1}{4}$ by $\frac{x (x - 5)}{x (x - 5)}$ .

$\frac{90 ((x - 5) \cdot 4)}{x ((x - 5) \cdot 4)} - \frac{90 (x \cdot 4)}{(x - 5) (x \cdot 4)} + \frac{x (x - 5)}{4 (x (x - 5))} = 0$

Step 2.1.7

Reorder the factors of $x ((x - 5) \cdot 4)$ .

$\frac{90 ((x - 5) \cdot 4)}{4 x (x - 5)} - \frac{90 (x \cdot 4)}{(x - 5) (x \cdot 4)} + \frac{x (x - 5)}{4 (x (x - 5))} = 0$

Step 2.1.8

Reorder the factors of $(x - 5) (x \cdot 4)$ .

$\frac{90 ((x - 5) \cdot 4)}{4 x (x - 5)} - \frac{90 (x \cdot 4)}{4 x (x - 5)} + \frac{x (x - 5)}{4 (x (x - 5))} = 0$

Step 2.1.9

Reorder the factors of $4 (x (x - 5))$ .

$\frac{90 ((x - 5) \cdot 4)}{4 x (x - 5)} - \frac{90 (x \cdot 4)}{4 x (x - 5)} + \frac{x (x - 5)}{4 x (x - 5)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{90 ((x - 5) \cdot 4) - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{90 (x \cdot 4 - 5 \cdot 4) - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.2

Move $4$ to the left of $x$ .

$\frac{90 (4 \cdot x - 5 \cdot 4) - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.3

Multiply $- 5$ by $4$ .

$\frac{90 (4 \cdot x - 20) - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{90 (4 x) + 90 \cdot - 20 - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.5

Multiply $4$ by $90$ .

$\frac{360 x + 90 \cdot - 20 - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.6

Multiply $90$ by $- 20$ .

$\frac{360 x - 1800 - 90 (x \cdot 4) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.7

Move $4$ to the left of $x$ .

$\frac{360 x - 1800 - 90 (4 \cdot x) + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.8

Multiply $4$ by $- 90$ .

$\frac{360 x - 1800 - 360 x + x (x - 5)}{4 x (x - 5)} = 0$

Step 2.3.9

Apply the distributive property.

$\frac{360 x - 1800 - 360 x + x \cdot x + x \cdot - 5}{4 x (x - 5)} = 0$

Step 2.3.10

Multiply $x$ by $x$ .

$\frac{360 x - 1800 - 360 x + x^{2} + x \cdot - 5}{4 x (x - 5)} = 0$

Step 2.3.11

Move $- 5$ to the left of $x$ .

$\frac{360 x - 1800 - 360 x + x^{2} - 5 x}{4 x (x - 5)} = 0$

Step 2.4

Combine the opposite terms in $360 x - 1800 - 360 x + x^{2} - 5 x$ .

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Step 2.4.1

Subtract $360 x$ from $360 x$ .

$\frac{0 - 1800 + x^{2} - 5 x}{4 x (x - 5)} = 0$

Step 2.4.2

Subtract $1800$ from $0$ .

$\frac{- 1800 + x^{2} - 5 x}{4 x (x - 5)} = 0$

Step 2.5

Factor $- 1800 + x^{2} - 5 x$ using the AC method.

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Step 2.5.1

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 $- 1800$ and whose sum is $- 5$ .

$- 45, 40$

Step 2.5.2

Write the factored form using these integers.

$\frac{(x - 45) (x + 40)}{4 x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 45) (x + 40) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

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

$x - 45 = 0$

$x + 40 = 0$

Step 4.2

Set $x - 45$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 45$ equal to $0$ .

$x - 45 = 0$

Step 4.2.2

Add $45$ to both sides of the equation.

$x = 45$

Step 4.3

Set $x + 40$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 40$ equal to $0$ .

$x + 40 = 0$

Step 4.3.2

Subtract $40$ from both sides of the equation.

$x = - 40$

Step 4.4

The final solution is all the values that make $(x - 45) (x + 40) = 0$ true.

$x = 45, - 40$