Home page for accesible maths 4 Chapter 4 contents 4.11 Discriminant test 4.13 Proof of the Discriminant Test

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4.12 Lemma on quadratic forms

Lemma.

Let $Q(h,k)=Ah^{2}+2Bhk+Ck^{2}$ and $\Delta=AC-B^{2}$ .

(i) If $A>0$ and $\Delta>0$ , then $Q(h,k)>0$ holds for all $(h,k)\neq 0$ ;

(ii) if $A<0$ and $\Delta>0$ , then $Q(h,k)<0$ holds for all $(h,k)\neq 0$ ;

(iii) if $\Delta<0$ , then $Q$ takes positive and negative values.

Brief proof. We have $Q(h,k)=k^{2}.\left(A\left(\frac{h}{k}\right)^{2}+2B\frac{h}{k}+C\right).$ Using the formula for the roots of a quadratic polynomial, we see that if $(2B)^{2}-4AC=-4\Delta<0$ then the quadratic polynomial $Ax^{2}+2Bx+C$ has no real roots, so is either always negative (if $A<0$ ) or always positive (if $A>0$ ). Setting $x=\frac{h}{k}$ and multiplying by $k^{2}$ , this proves (i) and (ii).

Finally, if $\Delta<0$ (and $A\neq 0$ ) then $Ax^{2}+2Bx+C$ has two distinct real roots, and so can have positive or negative values. Therefore so can $Q(h,k)$ .