Key to "Let f(x)=x2+ax+c where a,c are real numbers. Prove..."

mwombenizhjb

Answered question

2022-03-21

Let

f (x) = x^{2} + a x + c

where a,c are real numbers. Prove that there exist quadratic polynomials p(x) and q(x) (with real coefficients) having all roots real such that

f (x) = d \frac{1}{2} [p (x) + q (x)] .

Adan Berry

Beginner2022-03-22Added 12 answers

Without loss of generality by changing x into

x + \frac{a}{2}

, we can suppose that

a = 0

and write

f (x) = x^{2} + c

.
Now take

p (x) = x^{2} + B x + c

and

q (x) = x^{2} - B x + c

with

B^{2} > 4 c

. This ensures that the discriminants of p,q are positive and that p,q have real roots.

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