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If a > b are two real numbers such that a + b = 10 and a2 + b2 = 52, then what is the value of a - b ?
Explanation
To find the value of a - b, we use standard algebraic identities. Given:
1) a + b = 10
2) a2 + b2 = 52
First, find the value of 2ab using the identity (a + b)2 = a2 + b2 + 2ab:
102 = 52 + 2ab
100 = 52 + 2ab
2ab = 100 - 52 = 48
Next, use the identity (a - b)2 = a2 + b2 - 2ab:
(a - b)2 = 52 - 48
(a - b)2 = 4
Taking the square root, we get a - b = ±2. Since the problem specifies that a > b, the result must be positive. Therefore, a - b = 2.
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