Precalculus Examples

$\frac{x^{2} - 144}{19} = 12 + x$

Step 1

Multiply both sides by $19$ .

$\frac{x^{2} - 144}{19} \cdot 19 = (12 + x) \cdot 19$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{x^{2} - 144}{19} \cdot 19$ .

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Step 2.1.1.1

Simplify the numerator.

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Step 2.1.1.1.1

Rewrite $144$ as $12^{2}$ .

$\frac{x^{2} - 12^{2}}{19} \cdot 19 = (12 + x) \cdot 19$

Step 2.1.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 12$ .

$\frac{(x + 12) (x - 12)}{19} \cdot 19 = (12 + x) \cdot 19$

Step 2.1.1.2

Cancel the common factor of $19$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{(x + 12) (x - 12)}{19} \cdot 19 = (12 + x) \cdot 19$

Step 2.1.1.2.2

Rewrite the expression.

$(x + 12) (x - 12) = (12 + x) \cdot 19$

Step 2.1.1.3

Expand $(x + 12) (x - 12)$ using the FOIL Method.

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Step 2.1.1.3.1

Apply the distributive property.

$x (x - 12) + 12 (x - 12) = (12 + x) \cdot 19$

Step 2.1.1.3.2

Apply the distributive property.

$x \cdot x + x \cdot - 12 + 12 (x - 12) = (12 + x) \cdot 19$

Step 2.1.1.3.3

Apply the distributive property.

$x \cdot x + x \cdot - 12 + 12 x + 12 \cdot - 12 = (12 + x) \cdot 19$

Step 2.1.1.4

Simplify terms.

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Step 2.1.1.4.1

Combine the opposite terms in $x \cdot x + x \cdot - 12 + 12 x + 12 \cdot - 12$ .

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Step 2.1.1.4.1.1

Reorder the factors in the terms $x \cdot - 12$ and $12 x$ .

$x \cdot x - 12 x + 12 x + 12 \cdot - 12 = (12 + x) \cdot 19$

Step 2.1.1.4.1.2

Add $- 12 x$ and $12 x$ .

$x \cdot x + 0 + 12 \cdot - 12 = (12 + x) \cdot 19$

Step 2.1.1.4.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x + 12 \cdot - 12 = (12 + x) \cdot 19$

Step 2.1.1.4.2

Simplify each term.

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Step 2.1.1.4.2.1

Multiply $x$ by $x$ .

$x^{2} + 12 \cdot - 12 = (12 + x) \cdot 19$

Step 2.1.1.4.2.2

Multiply $12$ by $- 12$ .

$x^{2} - 144 = (12 + x) \cdot 19$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(12 + x) \cdot 19$ .

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Step 2.2.1.1

Apply the distributive property.

$x^{2} - 144 = 12 \cdot 19 + x \cdot 19$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Multiply $12$ by $19$ .

$x^{2} - 144 = 228 + x \cdot 19$

Step 2.2.1.2.2

Move $19$ to the left of $x$ .

$x^{2} - 144 = 228 + 19 \cdot x$

Step 2.2.1.2.3

Reorder $228$ and $19 x$ .

$x^{2} - 144 = 19 x + 228$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $19 x$ from both sides of the equation.

$x^{2} - 144 - 19 x = 228$

Step 3.2

Subtract $228$ from both sides of the equation.

$x^{2} - 144 - 19 x - 228 = 0$

Step 3.3

Subtract $228$ from $- 144$ .

$x^{2} - 19 x - 372 = 0$

Step 3.4

Factor $x^{2} - 19 x - 372$ using the AC method.

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Step 3.4.1

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 $- 372$ and whose sum is $- 19$ .

$- 31, 12$

Step 3.4.2

Write the factored form using these integers.

$(x - 31) (x + 12) = 0$

Step 3.5

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

$x - 31 = 0$

$x + 12 = 0$

Step 3.6

Set $x - 31$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x - 31$ equal to $0$ .

$x - 31 = 0$

Step 3.6.2

Add $31$ to both sides of the equation.

$x = 31$

Step 3.7

Set $x + 12$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Step 3.7.2

Subtract $12$ from both sides of the equation.

$x = - 12$

Step 3.8

The final solution is all the values that make $(x - 31) (x + 12) = 0$ true.

$x = 31, - 12$