Step 1: Define the numbers
Let the two numbers be \( x \) and \( y \).
Step 2: Use the given conditions
\[ x + y = 42 \] \[ xy = 437 \]
Step 3: Apply the identity for the difference of squares
\[ (x - y)^2 = (x + y)^2 - 4xy \]
Step 4: Substitute the given values
\[ (x - y)^2 = 42^2 - 4 \times 437 \] \[ = 1764 - 1748 = 16 \]
Step 5: Solve for \( x - y \)
\[ x - y = \sqrt{16} = 4 \]
Thus, the absolute difference between the numbers is 4.
If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \( (\alpha, \beta) \), then \( \beta - 2\alpha \) is equal to
If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to .