Finite Math Examples

$x^{2} + {(x + 4)}^{2} = {(x + 8)}^{2}$

Step 1

Subtract ${(x + 8)}^{2}$ from both sides of the equation.

$x^{2} + {(x + 4)}^{2} - {(x + 8)}^{2} = 0$

Step 2

Simplify $x^{2} + {(x + 4)}^{2} - {(x + 8)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$x^{2} + (x + 4) (x + 4) - {(x + 8)}^{2} = 0$

Step 2.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x^{2} + x (x + 4) + 4 (x + 4) - {(x + 8)}^{2} = 0$

Step 2.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 4 + 4 (x + 4) - {(x + 8)}^{2} = 0$

Step 2.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 - {(x + 8)}^{2} = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 - {(x + 8)}^{2} = 0$

Step 2.1.3.1.2

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

$x^{2} + x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 - {(x + 8)}^{2} = 0$

Step 2.1.3.1.3

Multiply $4$ by $4$ .

$x^{2} + x^{2} + 4 x + 4 x + 16 - {(x + 8)}^{2} = 0$

Step 2.1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + x^{2} + 8 x + 16 - {(x + 8)}^{2} = 0$

Step 2.1.4

Rewrite ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$x^{2} + x^{2} + 8 x + 16 - ((x + 8) (x + 8)) = 0$

Step 2.1.5

Expand $(x + 8) (x + 8)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$x^{2} + x^{2} + 8 x + 16 - (x (x + 8) + 8 (x + 8)) = 0$

Step 2.1.5.2

Apply the distributive property.

$x^{2} + x^{2} + 8 x + 16 - (x \cdot x + x \cdot 8 + 8 (x + 8)) = 0$

Step 2.1.5.3

Apply the distributive property.

$x^{2} + x^{2} + 8 x + 16 - (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8) = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + 8 x + 16 - (x^{2} + x \cdot 8 + 8 x + 8 \cdot 8) = 0$

Step 2.1.6.1.2

Move $8$ to the left of $x$ .

$x^{2} + x^{2} + 8 x + 16 - (x^{2} + 8 \cdot x + 8 x + 8 \cdot 8) = 0$

Step 2.1.6.1.3

Multiply $8$ by $8$ .

$x^{2} + x^{2} + 8 x + 16 - (x^{2} + 8 x + 8 x + 64) = 0$

Step 2.1.6.2

Add $8 x$ and $8 x$ .

$x^{2} + x^{2} + 8 x + 16 - (x^{2} + 16 x + 64) = 0$

Step 2.1.7

Apply the distributive property.

$x^{2} + x^{2} + 8 x + 16 - x^{2} - (16 x) - 1 \cdot 64 = 0$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $16$ by $- 1$ .

$x^{2} + x^{2} + 8 x + 16 - x^{2} - 16 x - 1 \cdot 64 = 0$

Step 2.1.8.2

Multiply $- 1$ by $64$ .

$x^{2} + x^{2} + 8 x + 16 - x^{2} - 16 x - 64 = 0$

Step 2.2

Combine the opposite terms in $x^{2} + x^{2} + 8 x + 16 - x^{2} - 16 x - 64$ .

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Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$x^{2} + 8 x + 16 + 0 - 16 x - 64 = 0$

Step 2.2.2

Add $x^{2} + 8 x + 16$ and $0$ .

$x^{2} + 8 x + 16 - 16 x - 64 = 0$

Step 2.3

Subtract $16 x$ from $8 x$ .

$x^{2} - 8 x + 16 - 64 = 0$

Step 2.4

Subtract $64$ from $16$ .

$x^{2} - 8 x - 48 = 0$

Step 3

Factor $x^{2} - 8 x - 48$ using the AC method.

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Step 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 $- 48$ and whose sum is $- 8$ .

$- 12, 4$

Step 3.2

Write the factored form using these integers.

$(x - 12) (x + 4) = 0$

Step 4

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

$x - 12 = 0$

$x + 4 = 0$

Step 5

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

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Step 5.1

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

$x - 12 = 0$

Step 5.2

Add $12$ to both sides of the equation.

$x = 12$

Step 6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $(x - 12) (x + 4) = 0$ true.

$x = 12, - 4$