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If the two roots of the quadratic equation \(x^2+kx+16=0\) are equal, what is the value of \(k^2\)?यदि द्विघात समीकरण \(x^2+kx+16=0\) के दोनों मूल समान हैं, तो \(k^2\) का मान क्या होगा?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: 6464
For equal roots, the discriminant must be zero. Here, \(a=1\), \(b=k\), and \(c=16\), so \(D=b^2-4ac=k^2-4(1)(16)=k^2-64\). Thus, \(k^2-64=0\), giving \(k^2=64\). Exam tip: For equal roots, immediately apply \(b^2-4ac=0\); do not confuse the constant term 16 with the required value of \(k^2\).समान मूलों के लिए विभेदक \(D=0\) होता है। यहाँ \(a=1\), \(b=k\) और \(c=16\) हैं, इसलिए \(D=b^2-4ac=k^2-4(1)(16)=k^2-64\)। अतः \(k^2-64=0\), जिससे \(k^2=64\)। परीक्षा-युक्ति: समान मूलों वाले द्विघात समीकरण में सीधे \(b^2-4ac=0\) लगाएँ; केवल स्थिर पद 16 को उत्तर न मानें।
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