Find the Roots (Zeros) (x+5)^2+(x-2)^2=37

${(x + 5)}^{2} + {(x - 2)}^{2} = 37$

Subtract $37$ from both sides of the equation.

${(x + 5)}^{2} + {(x - 2)}^{2} - 37 = 0$

Simplify ${(x + 5)}^{2} + {(x - 2)}^{2} - 37$ .

Simplify each term.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$(x + 5) (x + 5) + {(x - 2)}^{2} - 37 = 0$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$x (x + 5) + 5 (x + 5) + {(x - 2)}^{2} - 37 = 0$

Apply the distributive property.

$x \cdot x + x \cdot 5 + 5 (x + 5) + {(x - 2)}^{2} - 37 = 0$

Apply the distributive property.

$x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 + {(x - 2)}^{2} - 37 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 5 + 5 x + 5 \cdot 5 + {(x - 2)}^{2} - 37 = 0$

Move $5$ to the left of $x$ .

$x^{2} + 5 \cdot x + 5 x + 5 \cdot 5 + {(x - 2)}^{2} - 37 = 0$

Multiply $5$ by $5$ .

$x^{2} + 5 x + 5 x + 25 + {(x - 2)}^{2} - 37 = 0$

Add $5 x$ and $5 x$ .

$x^{2} + 10 x + 25 + {(x - 2)}^{2} - 37 = 0$

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

$x^{2} + 10 x + 25 + (x - 2) (x - 2) - 37 = 0$

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

Apply the distributive property.

$x^{2} + 10 x + 25 + x (x - 2) - 2 (x - 2) - 37 = 0$

Apply the distributive property.

$x^{2} + 10 x + 25 + x \cdot x + x \cdot - 2 - 2 (x - 2) - 37 = 0$

Apply the distributive property.

$x^{2} + 10 x + 25 + x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 37 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 10 x + 25 + x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 37 = 0$

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

$x^{2} + 10 x + 25 + x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 37 = 0$

Multiply $- 2$ by $- 2$ .

$x^{2} + 10 x + 25 + x^{2} - 2 x - 2 x + 4 - 37 = 0$

Subtract $2 x$ from $- 2 x$ .

$x^{2} + 10 x + 25 + x^{2} - 4 x + 4 - 37 = 0$

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

$2 x^{2} + 10 x + 25 - 4 x + 4 - 37 = 0$

Subtract $4 x$ from $10 x$ .

$2 x^{2} + 6 x + 25 + 4 - 37 = 0$

Add $25$ and $4$ .

$2 x^{2} + 6 x + 29 - 37 = 0$

Subtract $37$ from $29$ .

$2 x^{2} + 6 x - 8 = 0$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 6 x - 8$ .

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

$2 (x^{2}) + 6 x - 8 = 0$

Factor $2$ out of $6 x$ .

$2 (x^{2}) + 2 (3 x) - 8 = 0$

Factor $2$ out of $- 8$ .

$2 x^{2} + 2 (3 x) + 2 \cdot - 4 = 0$

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

$2 (x^{2} + 3 x) + 2 \cdot - 4 = 0$

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 4$ .

$2 (x^{2} + 3 x - 4) = 0$

Factor.

Factor $x^{2} + 3 x - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Write the factored form using these integers.

$2 ((x - 1) (x + 4)) = 0$

Remove unnecessary parentheses.

$2 (x - 1) (x + 4) = 0$

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

$x - 1 = 0$

$x + 4 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $2 (x - 1) (x + 4) = 0$ true.

$x = 1, - 4$

Find the Roots (Zeros) (x+5)^2+(x-2)^2=37

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