Solve by Factoring x^2+(x+1)^2=613

$x^{2} + {(x + 1)}^{2} = 613$

Subtract $613$ from both sides of the equation.

$x^{2} + {(x + 1)}^{2} - 613 = 0$

Simplify $x^{2} + {(x + 1)}^{2} - 613$ .

Simplify each term.

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

$x^{2} + (x + 1) (x + 1) - 613 = 0$

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

Apply the distributive property.

$x^{2} + x (x + 1) + 1 (x + 1) - 613 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 1 + 1 (x + 1) - 613 = 0$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - 613 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 - 613 = 0$

Multiply $x$ by $1$ .

$x^{2} + x^{2} + x + 1 x + 1 \cdot 1 - 613 = 0$

Multiply $x$ by $1$ .

$x^{2} + x^{2} + x + x + 1 \cdot 1 - 613 = 0$

Multiply $1$ by $1$ .

$x^{2} + x^{2} + x + x + 1 - 613 = 0$

Add $x$ and $x$ .

$x^{2} + x^{2} + 2 x + 1 - 613 = 0$

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

$2 x^{2} + 2 x + 1 - 613 = 0$

Subtract $613$ from $1$ .

$2 x^{2} + 2 x - 612 = 0$

Factor $2$ out of $2 x^{2} + 2 x - 612$ .

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

$2 (x^{2}) + 2 x - 612 = 0$

Factor $2$ out of $2 x$ .

$2 (x^{2}) + 2 (x) - 612 = 0$

Factor $2$ out of $- 612$ .

$2 (x^{2}) + 2 x + 2 \cdot - 306 = 0$

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

$2 (x^{2} + x) + 2 \cdot - 306 = 0$

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 306$ .

$2 (x^{2} + x - 306) = 0$

Factor.

Factor $x^{2} + x - 306$ 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 $- 306$ and whose sum is $1$ .

$- 17, 18$

Write the factored form using these integers.

$2 ((x - 17) (x + 18)) = 0$

Remove unnecessary parentheses.

$2 (x - 17) (x + 18) = 0$

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

$x - 17 = 0$

$x + 18 = 0$

Set $x - 17$ equal to $0$ and solve for $x$ .

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Add $17$ to both sides of the equation.

$x = 17$

Set $x + 18$ equal to $0$ and solve for $x$ .

Set $x + 18$ equal to $0$ .

$x + 18 = 0$

Subtract $18$ from both sides of the equation.

$x = - 18$

The final solution is all the values that make $2 (x - 17) (x + 18) = 0$ true.

$x = 17, - 18$

Solve by Factoring x^2+(x+1)^2=613

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