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Square roots are fundamental concepts in mathematics. When you take the square root of a number, you are finding a value which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16. Square roots are denoted using the radical symbol \( \sqrt{} \).In the context of this problem, square roots are used to solve the equation \( 16 = (3 - b)^2 + 9 \). To find the values of \( b \), we need to eliminate the square root by squaring both sides of the equation. This process allows us to solve for unknowns more easily by removing the radical sign. It's also important to remember that there are always two possible square roots for a positive number: one positive and one negative, which is why in our step-by-step solution, we have two scenarios for \( b \) - \( b = 3 - \sqrt{7} \) and \( b = 3 + \sqrt{7} \). This shows how square roots can lead to multiple solutions in algebraic equations.

Algebraic equations are equations that involve variables and can often include coefficients and constants. Solving these equations means finding the values of the variables which satisfy the equation. In our exercise, the main algebraic equation we consider is \( 16 = (3 - b)^2 + 9 \). This equation stems from the distance formula and incorporates squared terms, requiring us to isolate \( b \). - Start by rearranging: Subtract static numbers from both sides, so we have \( 7 = (3 - b)^2 \).- Solve by extracting square roots: Isolate the squared variable and find \( b \). Remember that working through such equations often requires multiple operations. From isolating the variable to factoring, these processes are essential core skills in algebra that enable one to derive the final values of variables, as shown in our solutions: \( 3 - \sqrt{7} \) and \( 3 + \sqrt{7} \).

Coordinate geometry is a branch of geometry where we use the coordinate plane to define points and shapes. This type of geometry uses algebra to analyze geometric problems.The distance formula, \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), is fundamental in coordinate geometry. It helps determine the distance between any two points in the plane, using their coordinates. In this exercise, we find the distance between point \((b, -1)\) and point \((3, 2)\).- Substitute coordinates into the formula.- Solve step by step to derive the equation with respect to \( b \).By applying the distance formula, one can analyze the spatial relationships of geometric figures. Moreover, employing these calculations facilitates a deeper understanding of how algebra can represent geometric concepts clearly. In this task, understanding the interplay between algebra and coordinate geometry is key to finding multiple values that satisfy the given conditions.