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Exercise 2206Linear algebra › Quadratic formsL1Added 3 Nov 2026

An inequality via a positive definite form

#inequality#positive definite#quadratic form

Statement

Show that for all x,y∈Rx,y\in\mathbb{R},
x2+xy+y2≥34(x+y)2.x^2+xy+y^2\ge\frac34(x+y)^2.

Hint

Consider the quadratic form Q=x2+xy+y2−34(x+y)2Q=x^2+xy+y^2-\frac34(x+y)^2 and show it is positive semi-definite.

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Solution

Compute QQ.
Q=x2+xy+y2−34(x2+2xy+y2)=14x2−12xy+14y2=14(x−y)2.Q=x^2+xy+y^2-\frac34(x^2+2xy+y^2)=\frac14x^2-\frac12xy+\frac14y^2=\frac14(x-y)^2.

Conclusion. Q=14(x−y)2≥0Q=\frac14(x-y)^2\ge0, with equality iff x=yx=y. Hence the inequality holds.

Remark. This is a special case of the Cauchy-Schwarz-type inequalities involving quadratic forms.

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