If the roots of the equation \(x^2+px+18=0\) are in the ratio \(1:2\) and both roots are negative, what is the value of \(p\)?
Answer and explanation
Correct answer: 9
Since both roots are negative and their ratio is \(1:2\), let them be \(-t\) and \(-2t\). Their product is \((-t)(-2t)=2t^2=18\), giving \(t=3\). Thus, the roots are \(-3\) and \(-6\), whose sum is \(-9\). By Vieta’s relation, the sum of the roots is \(-p\), so \(-p=-9\) and \(p=9\). Exam tip: for \(x^2+px+c=0\), the sum of the roots is \(-p\).
Frequently asked questions
What is the correct answer to this question?
9
Why is this the correct answer?
Since both roots are negative and their ratio is \(1:2\), let them be \(-t\) and \(-2t\). Their product is \((-t)(-2t)=2t^2=18\), giving \(t=3\). Thus, the roots are \(-3\) and \(-6\), whose sum is \(-9\). By Vieta’s relation, the sum of the roots is \(-p\), so \(-p=-9\) and \(p=9\). Exam tip: for \(x^2+px+c=0\), the sum of the roots is \(-p\).
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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