Let a, b, c be real. If ax 2 + bx + c = 0 has two real roots α and β , where α 1, then show that 1 + c/a + (modulus of b/a)

Question

Let a, b, c be real. If ax 2 + bx + c = 0 has two real roots &alpha; and &beta; , where &alpha; < -1 and &beta; > 1, then show that 1 + c/a + (modulus of b/a) < 0 .

mycroft holmes · Answer

WLOG we may assume b/a>0 [ otherwise consider the quadratic with roots -α, and -β, notice the conditions remain identical]

Now, 1 + b/a + c/a = 1 + α+β+αβ = (1+α)(1+β).

Now α<-1, so 1 + α<0. Also β>1, so 1+β>0. Hence (1+α)(1+β)<0

Hence 1 + b/a + c/a<0

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Let a, b, c be real. If ax 2 + bx + c = 0 has two real roots α and β , where α β > 1, then show that 1 + c/a + (modulus of b/a)

Let a, b, c be real. If ax² + bx + c = 0 has two real roots α and β , where α < -1 and β > 1, then show that 1 + c/a + (modulus of b/a) < 0 .

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