Question
Download Solution PDFसमीकरण (5 + \(\sqrt{2}\))x2 - (4 + \(\sqrt{5}\))x + (8 + \(2\sqrt{5}\)) = 0 के मूलों का हरात्मक माध्य ____ है।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFप्रयुक्त सूत्र:
यदि कोई द्विघात समीकरण ax2 + bx + c = 0 है
तब,
मूलों का योग (α + β) = −b/a और,
मूलों का गुणनफल (αβ) = c/a
यदि द्विघात समीकरण जिसके दो मूल होंगे, तो दो संख्याओं α और β का हरात्मक माध्य (HM) निम्न द्वारा दिया जाता है,
HM = \(\frac{2\alpha \beta}{\alpha +\beta }\)
अनुप्रयोग:
हमारे पास द्विघात समीकरण निम्न है,
(5 + \(\sqrt{2}\))x2 - (4 + \(\sqrt{5}\))x + (8 + \(2\sqrt{5}\)) = 0
अतः,
a = (5 + \(\sqrt{2}\))
b = - (4 + \(\sqrt{5}\))
c = (8 + \(2\sqrt{5}\))
उपरोक्त सूत्र का उपयोग करते हुए,
HM = \(\frac{2\alpha \beta}{\alpha +\beta }\) .... (1)
हम जानते हैं कि,
(αβ) = c/a = \(\frac{8+2\sqrt 5}{5+\sqrt 2}\)
(α + β) = −b/a = \(\frac{4+\sqrt 5}{5+\sqrt 2}\)
योग और गुणनफल के मानों को समीकरण (1) में रखने पर,
\(\rm HM = \frac{2 \left( \frac{8 + 2 \sqrt 5}{5 + \sqrt 2} \right)}{\frac{4+\sqrt 5}{5 + \sqrt 2}}= \frac{2 ( 8 + 2 \sqrt 5)}{4 + \sqrt 5} = 2 (2) = 4\)
Last updated on Jun 17, 2025
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