To find the value of \( x \) given the points \( M(-4,3) \) and \( N(x,6) \) where the distance between them \( |MN| \) is \( 3\sqrt{10} \), we use the distance formula:
\[ |MN| = \sqrt{(x - (-4))^2 + (6 - 3)^2} \]
This simplifies to:
\[ |MN| = \sqrt{(x + 4)^2 + (3)^2} \]
Now, since \( |MN| = 3\sqrt{10} \), we set up the equation:
\[ \sqrt{(x + 4)^2 + 3^2} = 3\sqrt{10} \]
Squaring both sides, we eliminate the square root:
\[ (x + 4)^2 + 9 = (3\sqrt{10})^2 \]
Calculating \( (3\sqrt{10})^2 \):
\[ (3\sqrt{10})^2 = 9 \cdot 10 = 90 \]
Now we can rewrite the equation:
\[ (x + 4)^2 + 9 = 90 \]
Subtracting 9 from both sides gives:
\[ (x + 4)^2 = 81 \]
Taking the square root of both sides, we have:
\[ x + 4 = 9 \quad \text{or} \quad x + 4 = -9 \]
This leads to two possibilities for \( x \):
-
\( x + 4 = 9 \) \[ x = 9 - 4 = 5 \]
-
\( x + 4 = -9 \) \[ x = -9 - 4 = -13 \]
Thus, the two possible values of \( x \) are:
\[ \boxed{5} \quad \text{or} \quad \boxed{-13} \]