The sum of two positive numbers is 27 and their product is 176. The positive difference between them is:
5
Let the two numbers be \(a\) and \(b\), with \(a + b = 27\) and \(ab = 176\).
Using the identity \((a-b)^2 = (a+b)^2 - 4ab\):
\((a-b)^2 = 27^2 - 4 \times 176 = 729 - 704 = 25\).
\(a - b = \sqrt{25} = 5\).
Verification: the numbers are 16 and 11, since \(16 + 11 = 27\) and \(16 \times 11 = 176\).
Hence, the positive difference between the two numbers is 5.
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Important Questions from Algebric Equations
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
If m + n = 24, then (m - 16)³ + (n - 8)³ is ____.
For the following equations, what are the values of a and b to have infinitely many solutions?
ax + by = 2
3x - (5 - 2ay) = 6
Simplify the following expression:
\(\frac{(x - y)^3 + (y - z)^3 + (z - x)^3}{(x - y)(y - z)(z - x)}\)
