Question
There are two natural numbers such that the square of
the smaller number exceeds five times the larger number by 4. Additionally, the sum of these two numbers is 20. Find the product of these two numbers.Solution
Let the smaller number be 'a' and larger number be 'b'. a + b = 20 -------- (I) And, a2Â = 5b + 4 Or, a2Â = 5 X (20 - a) + 4 (from equation I) Or, a2Â = 100 - 5a + 4 Or, a2Â + 5a - 104 = 0 Or, a2Â + 13a - 8a - 104 = 0 Or, a(a + 13) - 8(a + 13) = 0 Or, (a - 8) (a + 13) = 0 So, 'a' = 8 or 'a' = - 13 But given that both are natural numbers, So, 'a' = 8 On putting value of 'a' in equation I, We get, 'b' = 20 - 8 = 12 Therefore, required product = 12 X 8 = 96
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