If \(\alpha,\beta\) are the roots of \(x^2-20x+96=0\), what is the value of \((\alpha-8)(\beta-8)\)?
Answer and explanation
Correct answer: 0
By Vieta's formulas for \(x^2-20x+96=0\), we have \(\alpha+\beta=20\) and \(\alpha\beta=96\). Thus
\((\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64=96-8\times20+64=96-160+64=0.\)
Notes on distractors: 96 is a common mistake from taking \(\alpha\beta\) directly; -64 can arise from a sign error when handling the constant term. Exam tip: Always compute sum and product of roots first (Vieta) and then substitute into the expanded expression to avoid algebraic slips.
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What is the correct answer to this question?
0
Why is this the correct answer?
By Vieta's formulas for \(x^2-20x+96=0\), we have \(\alpha+\beta=20\) and \(\alpha\beta=96\). Thus
\((\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64=96-8\times20+64=96-160+64=0.\)
Notes on distractors: 96 is a common mistake from taking \(\alpha\beta\) directly; -64 can arise from a sign error when handling the constant term. Exam tip: Always compute sum and product of roots first (Vieta) and then substitute into the expanded expression to avoid algebraic slips.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Introduction to Quadratic Equations.
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