If \(\alpha\) and \(\beta\) are the roots of \(x^2-18x+80=0\), what is the value of \((\alpha-8)(\beta-8)\)?
Answer and explanation
Correct answer: 0
Use the identity \((\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64\). For \(x^2-18x+80=0\), by Vieta \(\alpha+\beta=18\) and \(\alpha\beta=80\). Therefore the value is \(80-8\times18+64=80-144+64=0\). A common mistake is to pick \(80\) (the product \(\alpha\beta\)) and ignore the linear shift terms. Exam tip: Apply Vieta's formulas directly or set \(y=x-8\) to transform the expression quickly.
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What is the correct answer to this question?
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Why is this the correct answer?
Use the identity \((\alpha-8)(\beta-8)=\alpha\beta-8(\alpha+\beta)+64\). For \(x^2-18x+80=0\), by Vieta \(\alpha+\beta=18\) and \(\alpha\beta=80\). Therefore the value is \(80-8\times18+64=80-144+64=0\). A common mistake is to pick \(80\) (the product \(\alpha\beta\)) and ignore the linear shift terms. Exam tip: Apply Vieta's formulas directly or set \(y=x-8\) to transform the expression quickly.
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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