Precalculus Examples

$a - \frac{5}{a} = 4$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, a, 1$

Step 1.2

The LCM of one and any expression is the expression.

$a$

Step 2

Multiply each term in $a - \frac{5}{a} = 4$ by $a$ to eliminate the fractions.

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Step 2.1

Multiply each term in $a - \frac{5}{a} = 4$ by $a$ .

$a \cdot a - \frac{5}{a} a = 4 a$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $a$ by $a$ .

$a^{2} - \frac{5}{a} a = 4 a$

Step 2.2.1.2

Cancel the common factor of $a$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{5}{a}$ into the numerator.

$a^{2} + \frac{- 5}{a} a = 4 a$

Step 2.2.1.2.2

Cancel the common factor.

$a^{2} + \frac{- 5}{a} a = 4 a$

Step 2.2.1.2.3

Rewrite the expression.

$a^{2} - 5 = 4 a$

Step 3

Solve the equation.

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Step 3.1

Subtract $4 a$ from both sides of the equation.

$a^{2} - 5 - 4 a = 0$

Step 3.2

Factor $a^{2} - 5 - 4 a$ using the AC method.

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Step 3.2.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2.2

Write the factored form using these integers.

$(a - 5) (a + 1) = 0$

Step 3.3

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

$a - 5 = 0$

$a + 1 = 0$

Step 3.4

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

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Step 3.4.1

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

$a - 5 = 0$

Step 3.4.2

Add $5$ to both sides of the equation.

$a = 5$

Step 3.5

Set $a + 1$ equal to $0$ and solve for $a$ .

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Step 3.5.1

Set $a + 1$ equal to $0$ .

$a + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 3.6

The final solution is all the values that make $(a - 5) (a + 1) = 0$ true.

$a = 5, - 1$