Prove that two coefficients of a quadratic function

Hugh Soto

Hugh Soto

Answered question

2022-03-31

Prove that two coefficients of a quadratic function are always positive
Let f(x) be a quadratic function of the form x2+px+q such that the equation f(f(x))=0 has two equal real roots. Show that p0 and q0.

Answer & Explanation

davane6a7a

davane6a7a

Beginner2022-04-01Added 8 answers

Since f(f(x))=0 does have a solution, f(x) can be factored, into say f(x)=(xα)(xβ). Here, we have α+β=p,αβ=q.
If α=β, you can show α and β must be 0. In this case, we have p=q=0.
Now, assume αβ. Then, f(f(x))=(f(x)α)(f(x)β)=0 is equivalent to f(x)α=0 or f(x)β=0. Since both of these equations cannot be true at the same time (since αβ) and f(f(x))=0 has only one solution, it must be the case that one of the two equations has only one root (i.e., a double root), and the other no roots.
Without loss of generality, we can assume f(x)α=0 has a double root, and f(x)β=0 has no roots. Looking at the discriminant, we get (αβ)24(αβα)=0 and (αβ)24(αββ)<0.
Now, we have (αβ)24(αβα)=(αβ)2+4α=0, so α0. Similarly, we have β<0. Hence, p=αβ0, and q=αβ0

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