What is the correct condition on \(t\) for the equation \(x^2+2(t+1)x+(3t+7)=0\) to have no real roots?
Answer and explanation
Correct answer: \(-2<t<3\)
Here, \(a=1\), \(b=2(t+1)\), and \(c=3t+7\). Therefore, the discriminant is \(D=b^2-4ac=4(t+1)^2-4(3t+7)=4(t-3)(t+2)\). An equation has no real roots when \(D<0\), so \((t-3)(t+2)<0\), giving \(-2<t<3\). Option B is incorrect because it does not come from the zeros \(-2\) and \(3\); at the endpoints \(t=-2,3\), \(D=0\) and the equation has equal real roots. Exam tip: For a quadratic equation, use \(D<0\) to identify the condition for no real roots.
Frequently asked questions
What is the correct answer to this question?
\(-2<t<3\)
Why is this the correct answer?
Here, \(a=1\), \(b=2(t+1)\), and \(c=3t+7\). Therefore, the discriminant is \(D=b^2-4ac=4(t+1)^2-4(3t+7)=4(t-3)(t+2)\). An equation has no real roots when \(D<0\), so \((t-3)(t+2)<0\), giving \(-2<t<3\). Option B is incorrect because it does not come from the zeros \(-2\) and \(3\); at the endpoints \(t=-2,3\), \(D=0\) and the equation has equal real roots. Exam tip: For a quadratic equation, use \(D<0\) to identify the condition for no real roots.
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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