Which condition on \(m\) is necessary for the equation \(x^2+(m-2)x+1=0\) to have no real roots?
Answer and explanation
Correct answer: \(0<m<4\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=m-2\), and \(c=1\), so \(D=(m-2)^2-4\). Thus, \((m-2)^2<4\), which gives \(-2<m-2<2\), and hence \(0<m<4\). Therefore, option A is correct. In option C, \(D=0\), which gives two equal real roots rather than no real roots. Exam tip: For ‘no real roots’, first apply the condition \(D<0\).
Frequently asked questions
What is the correct answer to this question?
\(0<m<4\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=m-2\), and \(c=1\), so \(D=(m-2)^2-4\). Thus, \((m-2)^2<4\), which gives \(-2<m-2<2\), and hence \(0<m<4\). Therefore, option A is correct. In option C, \(D=0\), which gives two equal real roots rather than no real roots. Exam tip: For ‘no real roots’, first apply the condition \(D<0\).
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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