For the equation \(x^2-2x+(p+3)=0\) to have no real roots, what is the correct condition on \(p\)?
Answer and explanation
Correct answer: \(p>-2\)
A quadratic equation \(ax^2+bx+c=0\) has no real roots when its discriminant \(D=b^2-4ac\) is less than zero. Here, \(a=1\), \(b=-2\), and \(c=p+3\), so \(D=(-2)^2-4(1)(p+3)=-4(p+2)\). Thus, \(-4(p+2)<0\), which gives \(p>-2\). Remember that \(D=0\) gives one repeated real root, so \(p=-2\) is not included.
Frequently asked questions
What is the correct answer to this question?
\(p>-2\)
Why is this the correct answer?
A quadratic equation \(ax^2+bx+c=0\) has no real roots when its discriminant \(D=b^2-4ac\) is less than zero. Here, \(a=1\), \(b=-2\), and \(c=p+3\), so \(D=(-2)^2-4(1)(p+3)=-4(p+2)\). Thus, \(-4(p+2)<0\), which gives \(p>-2\). Remember that \(D=0\) gives one repeated real root, so \(p=-2\) is not included.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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