Solve for x x^2+(x+1)^2+(x+2)^2=1454

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

Subtract $1454$ from both sides of the equation.

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

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $x$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $x$ and $x$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

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

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

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

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

Add $2 x$ and $4 x$ .

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

Add $1$ and $4$ .

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

Subtract $1454$ from $5$ .

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

Factor the left side of the equation.

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

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

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

Factor $3$ out of $6 x$ .

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

Factor $3$ out of $- 1449$ .

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

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

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

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

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

Factor.

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

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 $2$ .

$- 21, 23$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x - 21 = 0$

$x + 23 = 0$

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

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

$x - 21 = 0$

Add $21$ to both sides of the equation.

$x = 21$

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

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

$x + 23 = 0$

Subtract $23$ from both sides of the equation.

$x = - 23$

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

$x = 21, - 23$

Solve for x x^2+(x+1)^2+(x+2)^2=1454

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