For the quadratic equation \(2x^2+(k-4)x+8=0\) to have no real roots, which condition on \(k\) is correct?
Answer and explanation
Correct answer: \(-4<k<12\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=2\), \(b=k-4\), and \(c=8\), so \(D=(k-4)^2-4(2)(8)=(k-4)^2-64\). Therefore, \((k-4)^2<64\), which gives \(-8<k-4<8\), and hence \(-4<k<12\). Thus, option A is correct. In option B, \(D>0\), giving two distinct real roots, while in option C, \(D=0\), giving equal real roots. Exam tip: For the nature of roots, first calculate the discriminant using \(D=b^2-4ac\).
Frequently asked questions
What is the correct answer to this question?
\(-4<k<12\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=2\), \(b=k-4\), and \(c=8\), so \(D=(k-4)^2-4(2)(8)=(k-4)^2-64\). Therefore, \((k-4)^2<64\), which gives \(-8<k-4<8\), and hence \(-4<k<12\). Thus, option A is correct. In option B, \(D>0\), giving two distinct real roots, while in option C, \(D=0\), giving equal real roots. Exam tip: For the nature of roots, first calculate the discriminant using \(D=b^2-4ac\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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