Finite Math Examples

$x^{2} + {(x + 7)}^{2} = 13^{2}$

Step 1

Raise $13$ to the power of $2$ .

$x^{2} + {(x + 7)}^{2} = 169$

Step 2

Subtract $169$ from both sides of the equation.

$x^{2} + {(x + 7)}^{2} - 169 = 0$

Step 3

Simplify $x^{2} + {(x + 7)}^{2} - 169$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$x^{2} + (x + 7) (x + 7) - 169 = 0$

Step 3.1.2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$x^{2} + x (x + 7) + 7 (x + 7) - 169 = 0$

Step 3.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 7 + 7 (x + 7) - 169 = 0$

Step 3.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 - 169 = 0$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 7 + 7 x + 7 \cdot 7 - 169 = 0$

Step 3.1.3.1.2

Move $7$ to the left of $x$ .

$x^{2} + x^{2} + 7 \cdot x + 7 x + 7 \cdot 7 - 169 = 0$

Step 3.1.3.1.3

Multiply $7$ by $7$ .

$x^{2} + x^{2} + 7 x + 7 x + 49 - 169 = 0$

Step 3.1.3.2

Add $7 x$ and $7 x$ .

$x^{2} + x^{2} + 14 x + 49 - 169 = 0$

Step 3.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 14 x + 49 - 169 = 0$

Step 3.3

Subtract $169$ from $49$ .

$2 x^{2} + 14 x - 120 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $2$ out of $2 x^{2} + 14 x - 120$ .

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Step 4.1.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) + 14 x - 120 = 0$

Step 4.1.2

Factor $2$ out of $14 x$ .

$2 (x^{2}) + 2 (7 x) - 120 = 0$

Step 4.1.3

Factor $2$ out of $- 120$ .

$2 x^{2} + 2 (7 x) + 2 \cdot - 60 = 0$

Step 4.1.4

Factor $2$ out of $2 x^{2} + 2 (7 x)$ .

$2 (x^{2} + 7 x) + 2 \cdot - 60 = 0$

Step 4.1.5

Factor $2$ out of $2 (x^{2} + 7 x) + 2 \cdot - 60$ .

$2 (x^{2} + 7 x - 60) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} + 7 x - 60$ using the AC method.

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Step 4.2.1.1

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 $- 60$ and whose sum is $7$ .

$- 5, 12$

Step 4.2.1.2

Write the factored form using these integers.

$2 ((x - 5) (x + 12)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$2 (x - 5) (x + 12) = 0$

Step 5

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

$x - 5 = 0$

$x + 12 = 0$

Step 6

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

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Step 6.1

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

$x - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$x = 5$

Step 7

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

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Step 7.1

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

$x + 12 = 0$

Step 7.2

Subtract $12$ from both sides of the equation.

$x = - 12$

Step 8

The final solution is all the values that make $2 (x - 5) (x + 12) = 0$ true.

$x = 5, - 12$