For the equation \((p+1)x^2-2(p+2)x+(p+4)=0\), where \(p\ne -1\), what condition on \(p\) ensures that the roots are real?
Answer and explanation
Correct answer: \(p\leq 0,\ p\ne -1\)
Here, \(a=p+1\), \(b=-2(p+2)\), and \(c=p+4\). Therefore, the discriminant is \(D=b^2-4ac=4(p+2)^2-4(p+1)(p+4)=-4p\). For real roots, \(D\geq 0\), which gives \(p\leq 0\). Since \(p=-1\) is excluded, the complete condition is \(p\leq 0,\ p\ne -1\). Option B is incorrect because at \(p=0\), \(D=0\), giving two equal real roots. Exam tip: For real roots, apply \(D\geq 0\) and also verify that the coefficient of \(x^2\) is non-zero.
Frequently asked questions
What is the correct answer to this question?
\(p\leq 0,\ p\ne -1\)
Why is this the correct answer?
Here, \(a=p+1\), \(b=-2(p+2)\), and \(c=p+4\). Therefore, the discriminant is \(D=b^2-4ac=4(p+2)^2-4(p+1)(p+4)=-4p\). For real roots, \(D\geq 0\), which gives \(p\leq 0\). Since \(p=-1\) is excluded, the complete condition is \(p\leq 0,\ p\ne -1\). Option B is incorrect because at \(p=0\), \(D=0\), giving two equal real roots. Exam tip: For real roots, apply \(D\geq 0\) and also verify that the coefficient of \(x^2\) is non-zero.
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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