If the equation \(x^2-2(a-1)x+(a^2+1)=0\) has no real roots, which condition on \(a\) is correct?
Answer and explanation
Correct answer: \(a>0\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(D=[-2(a-1)]^2-4(a^2+1)=4(a-1)^2-4(a^2+1)=-8a\). Therefore, \(-8a<0\), which gives \(a>0\). For \(a=0\), \(D=0\), so the equation has two real and equal roots. Exam tip: whenever ‘no real roots’ is stated, immediately apply the condition \(D<0\).
Frequently asked questions
What is the correct answer to this question?
\(a>0\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(D=[-2(a-1)]^2-4(a^2+1)=4(a-1)^2-4(a^2+1)=-8a\). Therefore, \(-8a<0\), which gives \(a>0\). For \(a=0\), \(D=0\), so the equation has two real and equal roots. Exam tip: whenever ‘no real roots’ is stated, immediately 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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