The roots of the equation \(x^2-4x+1=0\) are \(\alpha\) and \(\beta\). What is the value of \(\alpha+\beta+\alpha\beta\)?
Answer and explanation
Correct answer: 5
For a quadratic equation \(ax^2+bx+c=0\), the sum of the roots is \(-\frac{b}{a}\) and their product is \(\frac{c}{a}\). Here, \(a=1, b=-4, c=1\), so \(\alpha+\beta=4\) and \(\alpha\beta=1\). Therefore, \(\alpha+\beta+\alpha\beta=4+1=5\). Exam tip: Use the sum and product of roots directly instead of solving the quadratic equation.
Frequently asked questions
What is the correct answer to this question?
5
Why is this the correct answer?
For a quadratic equation \(ax^2+bx+c=0\), the sum of the roots is \(-\frac{b}{a}\) and their product is \(\frac{c}{a}\). Here, \(a=1, b=-4, c=1\), so \(\alpha+\beta=4\) and \(\alpha\beta=1\). Therefore, \(\alpha+\beta+\alpha\beta=4+1=5\). Exam tip: Use the sum and product of roots directly instead of solving the quadratic equation.
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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