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If a and b are positive numbers and each of the equations x2+ax+ 2b = 0 and x2+ 2bx +a = 0 has real roots, then the smallest possible value of (a + b) is _______.

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answer is 6.

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Detailed Solution

x2+ax+2b=0 D≥0⇒a2-8b≥0⇒a2≥8b x2+2bx+a=0 D≥0⇒4b2-4a≥0⇒b2≥a⇒b4≥a2≥8b b4≥8b⇒b3≥8⇒b≥2 a2≥16⇒a≥4 a+b≥6

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If a and b are positive numbers and each of the equations x2+ax+ 2b = 0 and x2+ 2bx +a = 0 has real roots, then the smallest possible value of (a + b) is _______.