Precalculus Examples

$\frac{x^{2}}{x + 100} = 50$

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.

$x + 100, 1$

Step 1.2

Remove parentheses.

$x + 100, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x + 100$

Step 2

Multiply each term in $\frac{x^{2}}{x + 100} = 50$ by $x + 100$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x^{2}}{x + 100} = 50$ by $x + 100$ .

$\frac{x^{2}}{x + 100} (x + 100) = 50 (x + 100)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x + 100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x^{2}}{x + 100} (x + 100) = 50 (x + 100)$

Step 2.2.1.2

Rewrite the expression.

$x^{2} = 50 (x + 100)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$x^{2} = 50 x + 50 \cdot 100$

Step 2.3.2

Multiply $50$ by $100$ .

$x^{2} = 50 x + 5000$

Step 3

Solve the equation.

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Step 3.1

Subtract $50 x$ from both sides of the equation.

$x^{2} - 50 x = 5000$

Step 3.2

Subtract $5000$ from both sides of the equation.

$x^{2} - 50 x - 5000 = 0$

Step 3.3

Factor $x^{2} - 50 x - 5000$ using the AC method.

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Step 3.3.1

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 $- 5000$ and whose sum is $- 50$ .

$- 100, 50$

Step 3.3.2

Write the factored form using these integers.

$(x - 100) (x + 50) = 0$

Step 3.4

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

$x - 100 = 0$

$x + 50 = 0$

Step 3.5

Set $x - 100$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 100$ equal to $0$ .

$x - 100 = 0$

Step 3.5.2

Add $100$ to both sides of the equation.

$x = 100$

Step 3.6

Set $x + 50$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 50$ equal to $0$ .

$x + 50 = 0$

Step 3.6.2

Subtract $50$ from both sides of the equation.

$x = - 50$

Step 3.7

The final solution is all the values that make $(x - 100) (x + 50) = 0$ true.

$x = 100, - 50$