Find the Roots (Zeros) (x+1)^2+(x-8)^2=45

${(x + 1)}^{2} + {(x - 8)}^{2} = 45$

Step 1

Subtract $45$ from both sides of the equation.

${(x + 1)}^{2} + {(x - 8)}^{2} - 45 = 0$

Step 2

Simplify ${(x + 1)}^{2} + {(x - 8)}^{2} - 45$ .

Simplify each term.

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

$(x + 1) (x + 1) + {(x - 8)}^{2} - 45 = 0$

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

Apply the distributive property.

$x (x + 1) + 1 (x + 1) + {(x - 8)}^{2} - 45 = 0$

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1) + {(x - 8)}^{2} - 45 = 0$

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 + {(x - 8)}^{2} - 45 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 + {(x - 8)}^{2} - 45 = 0$

Multiply $x$ by $1$ .

$x^{2} + x + 1 x + 1 \cdot 1 + {(x - 8)}^{2} - 45 = 0$

Multiply $x$ by $1$ .

$x^{2} + x + x + 1 \cdot 1 + {(x - 8)}^{2} - 45 = 0$

Multiply $1$ by $1$ .

$x^{2} + x + x + 1 + {(x - 8)}^{2} - 45 = 0$

Add $x$ and $x$ .

$x^{2} + 2 x + 1 + {(x - 8)}^{2} - 45 = 0$

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$x^{2} + 2 x + 1 + (x - 8) (x - 8) - 45 = 0$

Expand $(x - 8) (x - 8)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 2 x + 1 + x (x - 8) - 8 (x - 8) - 45 = 0$

Apply the distributive property.

$x^{2} + 2 x + 1 + x \cdot x + x \cdot - 8 - 8 (x - 8) - 45 = 0$

Apply the distributive property.

$x^{2} + 2 x + 1 + x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8 - 45 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 2 x + 1 + x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8 - 45 = 0$

Move $- 8$ to the left of $x$ .

$x^{2} + 2 x + 1 + x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8 - 45 = 0$

Multiply $- 8$ by $- 8$ .

$x^{2} + 2 x + 1 + x^{2} - 8 x - 8 x + 64 - 45 = 0$

Subtract $8 x$ from $- 8 x$ .

$x^{2} + 2 x + 1 + x^{2} - 16 x + 64 - 45 = 0$

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

$2 x^{2} + 2 x + 1 - 16 x + 64 - 45 = 0$

Subtract $16 x$ from $2 x$ .

$2 x^{2} - 14 x + 1 + 64 - 45 = 0$

Add $1$ and $64$ .

$2 x^{2} - 14 x + 65 - 45 = 0$

Subtract $45$ from $65$ .

$2 x^{2} - 14 x + 20 = 0$

Step 3

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} - 14 x + 20$ .

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

$2 (x^{2}) - 14 x + 20 = 0$

Factor $2$ out of $- 14 x$ .

$2 (x^{2}) + 2 (- 7 x) + 20 = 0$

Factor $2$ out of $20$ .

$2 x^{2} + 2 (- 7 x) + 2 \cdot 10 = 0$

Factor $2$ out of $2 x^{2} + 2 (- 7 x)$ .

$2 (x^{2} - 7 x) + 2 \cdot 10 = 0$

Factor $2$ out of $2 (x^{2} - 7 x) + 2 \cdot 10$ .

$2 (x^{2} - 7 x + 10) = 0$

Factor.

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$2 ((x - 5) (x - 2)) = 0$

Remove unnecessary parentheses.

$2 (x - 5) (x - 2) = 0$

Step 4

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

$x - 5 = 0$

$x - 2 = 0$

Step 5

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Step 6

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $2 (x - 5) (x - 2) = 0$ true.

$x = 5, 2$

Step 8

Find the Roots (Zeros) (x+1)^2+(x-8)^2=45

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