Let's solve the given equation step-by-step:
[tex]\[
\sqrt{61} = \sqrt{((-3 - a)^2 + (7 - 2)^2)}
\][/tex]
First, simplify the expression inside the square root on the right-hand side. Start by calculating the constant term inside the square root:
[tex]\[
7 - 2 = 5
\][/tex]
Therefore, the equation becomes:
[tex]\[
\sqrt{61} = \sqrt{((-3 - a)^2 + 5^2)}
\][/tex]
Next, let's square both sides of the equation to eliminate the square roots:
[tex]\[
61 = (-3 - a)^2 + 5^2
\][/tex]
Now we need to simplify the right-hand side further:
[tex]\[
5^2 = 25
\][/tex]
Thus, the equation simplifies to:
[tex]\[
61 = (-3 - a)^2 + 25
\][/tex]
Subtract 25 from both sides to isolate the square term:
[tex]\[
61 - 25 = (-3 - a)^2
\][/tex]
Simplify the left-hand side:
[tex]\[
36 = (-3 - a)^2
\][/tex]
Now, take the square root of both sides to solve for [tex]\(-3 - a\)[/tex]:
[tex]\[
\sqrt{36} = \pm 6
\][/tex]
This gives us two possible equations:
1. [tex]\(-3 - a = 6\)[/tex]
2. [tex]\(-3 - a = -6\)[/tex]
Let's solve each equation separately:
For the first equation:
[tex]\[
-3 - a = 6
\][/tex]
Add 3 to both sides:
[tex]\[
-a = 9
\][/tex]
Multiply both sides by -1:
[tex]\[
a = -9
\][/tex]
For the second equation:
[tex]\[
-3 - a = -6
\][/tex]
Add 3 to both sides:
[tex]\[
-a = -3
\][/tex]
Multiply both sides by -1:
[tex]\[
a = 3
\][/tex]
Therefore, the solutions for [tex]\( a \)[/tex] are:
[tex]\[
a = -9 \quad \text{and} \quad a = 3
\][/tex]
