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Class 10MathematicsQuadratic EquationsRoots of a Quadratic EquationHardLevel 31
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If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-2x-8=0\), what is the value of \(\left(\alpha+3\right)\left(\beta+3\right)\)?

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Answer and explanation

Correct answer: 7

By Vieta’s formulas, \(\alpha+\beta=2\) and \(\alpha\beta=-8\). Hence, \(\left(\alpha+3\right)\left(\beta+3\right)=\alpha\beta+3(\alpha+\beta)+9=-8+3(2)+9=7\). Therefore, the correct answer is 7. Exam tip: when the same number is added to both roots, expand the product and use the sum and product of roots; using only \(\alpha+\beta\) would be incomplete.

Related tags

Quadratic EquationsVieta FormulasRoots Of EquationsTransformed RootsAlgebraic Identities

Frequently asked questions

What is the correct answer to this question?

7

Why is this the correct answer?

By Vieta’s formulas, \(\alpha+\beta=2\) and \(\alpha\beta=-8\). Hence, \(\left(\alpha+3\right)\left(\beta+3\right)=\alpha\beta+3(\alpha+\beta)+9=-8+3(2)+9=7\). Therefore, the correct answer is 7. Exam tip: when the same number is added to both roots, expand the product and use the sum and product of roots; using only \(\alpha+\beta\) would be incomplete.

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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