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

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

Step 1

Subtract $45$ from both sides of the equation.

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

Step 2

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $t$ by $t$ .

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

Multiply $t$ by $1$ .

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

Multiply $t$ by $1$ .

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

Multiply $1$ by $1$ .

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

Add $t$ and $t$ .

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $t$ by $t$ .

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

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

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

Multiply $- 8$ by $- 8$ .

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

Subtract $8 t$ from $- 8 t$ .

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

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

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

Subtract $16 t$ from $2 t$ .

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

Add $1$ and $64$ .

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

Subtract $45$ from $65$ .

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

Step 3

Factor the left side of the equation.

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

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

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

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

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

Factor $2$ out of $20$ .

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

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

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

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

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

Factor.

Factor $t^{2} - 7 t + 10$ 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.

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

Remove unnecessary parentheses.

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

Step 4

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

$t - 5 = 0$

$t - 2 = 0$

Step 5

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

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

$t - 5 = 0$

Add $5$ to both sides of the equation.

$t = 5$

Step 6

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

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

$t - 2 = 0$

Add $2$ to both sides of the equation.

$t = 2$

Step 7

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

$t = 5, 2$

Step 8

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

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