Let f(x)=x^2+ax+c where a,c are real numbers. Prove that there

sexe4yo

sexe4yo

Answered question

2022-04-30

Let f(x)=x2+ax+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)=12[p(x)+q(x)].

Answer & Explanation

Isla Preston

Isla Preston

Beginner2022-05-01Added 12 answers

Without loss of generality by changing x into x+a2, we can suppose that a=0 and write f(x)=x2+c.
Now take p(x)=x2+Bx+c and q(x)=x2Bx+c with B2>4c. This ensures that the discriminants of p,q are positive and that p,q have real roots.

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