For (x^2+2x+c=0), the roots are real and their product is less than their sum. What is the correct condition on (c)?
Answer and explanation
Correct answer: (c<-2)
For \(x^2+2x+c=0\), the sum of the roots is \(-2\), and their product is \(c\). The statement says the product is less than the sum, so \(c<-2\). This condition is already stronger than the real-root requirement. Indeed, the discriminant is \(D=2^2-4c=4-4c\), which is positive whenever \(c<1\), and every value satisfying \(c<-2\) also satisfies that condition.
Thus the complete combined condition is \(c<-2\), which is option A. The inequality must be strict because “less than” does not allow equality. Option B, \(c\le1\), only guarantees real roots and does not guarantee that the product is less than the sum. Therefore the supplied answer and explanation are correct.
Frequently asked questions
What is the correct answer to this question?
(c<-2)
Why is this the correct answer?
For \(x^2+2x+c=0\), the sum of the roots is \(-2\), and their product is \(c\). The statement says the product is less than the sum, so \(c<-2\). This condition is already stronger than the real-root requirement. Indeed, the discriminant is \(D=2^2-4c=4-4c\), which is positive whenever \(c<1\), and every value satisfying \(c<-2\) also satisfies that condition.
Thus the complete combined condition is \(c<-2\), which is option A. The inequality must be strict because “less than” does not allow equality. Option B, \(c\le1\), only guarantees real roots and does not guarantee that the product is less than the sum. Therefore the supplied answer and explanation are correct.
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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