To find the value of \( x \) such that the distance between the points \( M(-4, 3) \) and \( N(x, 6) \) is \( 3\sqrt{10} \), we can use the distance formula:
\[ |MN| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the points \( M \) and \( N \):
\[ |MN| = \sqrt{(x - (-4))^2 + (6 - 3)^2} \]
Simplifying this, we get:
\[ |MN| = \sqrt{(x + 4)^2 + (6 - 3)^2} \] \[ |MN| = \sqrt{(x + 4)^2 + 3^2} \] \[ |MN| = \sqrt{(x + 4)^2 + 9} \]
We know that the magnitude of \( |MN| \) is given as \( 3\sqrt{10} \):
\[ \sqrt{(x + 4)^2 + 9} = 3\sqrt{10} \]
Next, we square both sides to eliminate the square root:
\[ (x + 4)^2 + 9 = (3\sqrt{10})^2 \] \[ (x + 4)^2 + 9 = 9 \times 10 \] \[ (x + 4)^2 + 9 = 90 \]
Subtracting 9 from both sides:
\[ (x + 4)^2 = 81 \]
Now, we take the square root of both sides:
\[ x + 4 = 9 \quad \text{or} \quad x + 4 = -9 \]
Solving these two equations:
-
From \( x + 4 = 9 \): \[ x = 9 - 4 = 5 \]
-
From \( x + 4 = -9 \): \[ x = -9 - 4 = -13 \]
Thus, the possible values for \( x \) are \( 5 \) and \( -13 \).
Therefore, the solutions are:
\[ \boxed{5} \quad \text{and} \quad \boxed{-13} \]