If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-5x+1=0\), what is the value of \(\alpha^2+\beta^2\)?
Answer and explanation
Correct answer: 23
By Vieta’s formulas, \(\alpha+\beta=5\) and \(\alpha\beta=1\). Hence, \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=5^2-2(1)=25-2=23\). Therefore, the correct answer is 23. In the exam, remember the identity \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\); using only \((\alpha+\beta)^2\) gives 25, which is incorrect.
Frequently asked questions
What is the correct answer to this question?
23
Why is this the correct answer?
By Vieta’s formulas, \(\alpha+\beta=5\) and \(\alpha\beta=1\). Hence, \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=5^2-2(1)=25-2=23\). Therefore, the correct answer is 23. In the exam, remember the identity \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\); using only \((\alpha+\beta)^2\) gives 25, which is incorrect.
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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