Algebra Examples

$x^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} = 1454$

Step 1

Subtract $1454$ from both sides of the equation.

$x^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} - 1454 = 0$

Step 2

Simplify $x^{2} + {(x + 1)}^{2} + {(x + 2)}^{2} - 1454$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$x^{2} + (x + 1) (x + 1) + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x^{2} + x (x + 1) + 1 (x + 1) + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 1 + 1 (x + 1) + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 + {(x + 2)}^{2} - 1454 = 0$

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 $x$ .

$x^{2} + x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.3.1.2

Multiply $x$ by $1$ .

$x^{2} + x^{2} + x + 1 x + 1 \cdot 1 + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.3.1.3

Multiply $x$ by $1$ .

$x^{2} + x^{2} + x + x + 1 \cdot 1 + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.3.1.4

Multiply $1$ by $1$ .

$x^{2} + x^{2} + x + x + 1 + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.3.2

Add $x$ and $x$ .

$x^{2} + x^{2} + 2 x + 1 + {(x + 2)}^{2} - 1454 = 0$

Step 2.1.4

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

$x^{2} + x^{2} + 2 x + 1 + (x + 2) (x + 2) - 1454 = 0$

Step 2.1.5

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

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Step 2.1.5.1

Apply the distributive property.

$x^{2} + x^{2} + 2 x + 1 + x (x + 2) + 2 (x + 2) - 1454 = 0$

Step 2.1.5.2

Apply the distributive property.

$x^{2} + x^{2} + 2 x + 1 + x \cdot x + x \cdot 2 + 2 (x + 2) - 1454 = 0$

Step 2.1.5.3

Apply the distributive property.

$x^{2} + x^{2} + 2 x + 1 + x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 - 1454 = 0$

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 $x$ .

$x^{2} + x^{2} + 2 x + 1 + x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 - 1454 = 0$

Step 2.1.6.1.2

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

$x^{2} + x^{2} + 2 x + 1 + x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 - 1454 = 0$

Step 2.1.6.1.3

Multiply $2$ by $2$ .

$x^{2} + x^{2} + 2 x + 1 + x^{2} + 2 x + 2 x + 4 - 1454 = 0$

Step 2.1.6.2

Add $2 x$ and $2 x$ .

$x^{2} + x^{2} + 2 x + 1 + x^{2} + 4 x + 4 - 1454 = 0$

Step 2.2

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

$2 x^{2} + 2 x + 1 + x^{2} + 4 x + 4 - 1454 = 0$

Step 2.3

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

$3 x^{2} + 2 x + 1 + 4 x + 4 - 1454 = 0$

Step 2.4

Add $2 x$ and $4 x$ .

$3 x^{2} + 6 x + 1 + 4 - 1454 = 0$

Step 2.5

Add $1$ and $4$ .

$3 x^{2} + 6 x + 5 - 1454 = 0$

Step 2.6

Subtract $1454$ from $5$ .

$3 x^{2} + 6 x - 1449 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $3$ out of $3 x^{2} + 6 x - 1449$ .

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Step 3.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 6 x - 1449 = 0$

Step 3.1.2

Factor $3$ out of $6 x$ .

$3 (x^{2}) + 3 (2 x) - 1449 = 0$

Step 3.1.3

Factor $3$ out of $- 1449$ .

$3 x^{2} + 3 (2 x) + 3 \cdot - 483 = 0$

Step 3.1.4

Factor $3$ out of $3 x^{2} + 3 (2 x)$ .

$3 (x^{2} + 2 x) + 3 \cdot - 483 = 0$

Step 3.1.5

Factor $3$ out of $3 (x^{2} + 2 x) + 3 \cdot - 483$ .

$3 (x^{2} + 2 x - 483) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} + 2 x - 483$ using the AC method.

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Step 3.2.1.1

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 $- 483$ and whose sum is $2$ .

$- 21, 23$

Step 3.2.1.2

Write the factored form using these integers.

$3 ((x - 21) (x + 23)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$3 (x - 21) (x + 23) = 0$

Step 4

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

$x - 21 = 0$

$x + 23 = 0$

Step 5

Set $x - 21$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 21$ equal to $0$ .

$x - 21 = 0$

Step 5.2

Add $21$ to both sides of the equation.

$x = 21$

Step 6

Set $x + 23$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 23$ equal to $0$ .

$x + 23 = 0$

Step 6.2

Subtract $23$ from both sides of the equation.

$x = - 23$

Step 7

The final solution is all the values that make $3 (x - 21) (x + 23) = 0$ true.

$x = 21, - 23$