| Summary: | In this paper, we find the solution of the following quadratic functional equation <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub></mfenced><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>Q</mi><mfenced separators="" open="(" close=")"><msub><mo>∑</mo><mrow><mi>j</mi><mo>≠</mo><mi>i</mi></mrow></msub><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>x</mi><mi>i</mi></msub></mfenced></mrow></semantics></math></inline-formula>, which is derived from the gravity of the <i>n</i> distinct vectors <inline-formula><math display="inline"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> in an inner product space, and prove that the stability results of the <inline-formula><math display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-quadratic mappings in <inline-formula><math display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-homogeneous property.
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