Pre-Algebra Examples

$x^{2} + {(x + 2)}^{2} = {(x + 4)}^{2} + 65$

Step 1

Simplify ${(x + 4)}^{2} + 65$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$x^{2} + {(x + 2)}^{2} = (x + 4) (x + 4) + 65$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$x^{2} + {(x + 2)}^{2} = x (x + 4) + 4 (x + 4) + 65$

Step 1.1.2.2

Apply the distributive property.

$x^{2} + {(x + 2)}^{2} = x \cdot x + x \cdot 4 + 4 (x + 4) + 65$

Step 1.1.2.3

Apply the distributive property.

$x^{2} + {(x + 2)}^{2} = x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4 + 65$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + {(x + 2)}^{2} = x^{2} + x \cdot 4 + 4 x + 4 \cdot 4 + 65$

Step 1.1.3.1.2

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

$x^{2} + {(x + 2)}^{2} = x^{2} + 4 \cdot x + 4 x + 4 \cdot 4 + 65$

Step 1.1.3.1.3

Multiply $4$ by $4$ .

$x^{2} + {(x + 2)}^{2} = x^{2} + 4 x + 4 x + 16 + 65$

Step 1.1.3.2

Add $4 x$ and $4 x$ .

$x^{2} + {(x + 2)}^{2} = x^{2} + 8 x + 16 + 65$

Step 1.2

Add $16$ and $65$ .

$x^{2} + {(x + 2)}^{2} = x^{2} + 8 x + 81$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} + {(x + 2)}^{2} - x^{2} = 8 x + 81$

Step 2.2

Subtract $8 x$ from both sides of the equation.

$x^{2} + {(x + 2)}^{2} - x^{2} - 8 x = 81$

Step 2.3

Combine the opposite terms in $x^{2} + {(x + 2)}^{2} - x^{2} - 8 x$ .

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Step 2.3.1

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

${(x + 2)}^{2} + 0 - 8 x = 81$

Step 2.3.2

Add ${(x + 2)}^{2}$ and $0$ .

${(x + 2)}^{2} - 8 x = 81$

Step 2.4

Simplify each term.

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Step 2.4.1

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

$(x + 2) (x + 2) - 8 x = 81$

Step 2.4.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.4.2.1

Apply the distributive property.

$x (x + 2) + 2 (x + 2) - 8 x = 81$

Step 2.4.2.2

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 (x + 2) - 8 x = 81$

Step 2.4.2.3

Apply the distributive property.

$x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - 8 x = 81$

Step 2.4.3

Simplify and combine like terms.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - 8 x = 81$

Step 2.4.3.1.2

Move $2$ to the left of $x$ .

$x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - 8 x = 81$

Step 2.4.3.1.3

Multiply $2$ by $2$ .

$x^{2} + 2 x + 2 x + 4 - 8 x = 81$

Step 2.4.3.2

Add $2 x$ and $2 x$ .

$x^{2} + 4 x + 4 - 8 x = 81$

Step 2.5

Subtract $8 x$ from $4 x$ .

$x^{2} - 4 x + 4 = 81$

Step 3

Subtract $81$ from both sides of the equation.

$x^{2} - 4 x + 4 - 81 = 0$

Step 4

Subtract $81$ from $4$ .

$x^{2} - 4 x - 77 = 0$

Step 5

Factor $x^{2} - 4 x - 77$ using the AC method.

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Step 5.1

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 $- 77$ and whose sum is $- 4$ .

$- 11, 7$

Step 5.2

Write the factored form using these integers.

$(x - 11) (x + 7) = 0$

Step 6

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

$x - 11 = 0$

$x + 7 = 0$

Step 7

Set $x - 11$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Step 7.2

Add $11$ to both sides of the equation.

$x = 11$

Step 8

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 8.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 9

The final solution is all the values that make $(x - 11) (x + 7) = 0$ true.

$x = 11, - 7$