If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-10x+21=0\), what is the value of \(\alpha^3+\beta^3\)?
Answer and explanation
Correct answer: 370
From the coefficients, \(\alpha+\beta=10\) and \(\alpha\beta=21\). Using the identity \(\alpha^3+\beta^3=(\alpha+eta)^3-3\alpha\beta(\alpha+eta)\), we get \(10^3-3\times21\times10=1000-630=370\). Hence, the correct answer is 370. Exam tip: For the sum of cubes of roots, first find their sum and product using Vieta’s relations.
Frequently asked questions
What is the correct answer to this question?
370
Why is this the correct answer?
From the coefficients, \(\alpha+\beta=10\) and \(\alpha\beta=21\). Using the identity \(\alpha^3+\beta^3=(\alpha+eta)^3-3\alpha\beta(\alpha+eta)\), we get \(10^3-3\times21\times10=1000-630=370\). Hence, the correct answer is 370. Exam tip: For the sum of cubes of roots, first find their sum and product using Vieta’s relations.
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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