If α and β are roots of x² − 6x − 27 = 0, what is αβ + 3α + 3β?
Answer and explanation
Correct answer: −9
For a quadratic x² + bx + c = 0, Vieta’s relations state that the sum of the roots is −b and their product is c. Comparing the given equation x² − 6x − 27 = 0 with this form gives α + β = 6 and αβ = −27. Rewrite the required expression by grouping the linear terms: αβ + 3α + 3β = αβ + 3(α + β). Substitution now gives −27 + 3(6) = −27 + 18 = −9. Therefore option A is correct. Option B is only the product αβ and ignores the remaining terms. Option C has the wrong sign, while option D results from an incorrect use of the root sum and product. No individual roots need to be calculated.
Frequently asked questions
What is the correct answer to this question?
−9
Why is this the correct answer?
For a quadratic x² + bx + c = 0, Vieta’s relations state that the sum of the roots is −b and their product is c. Comparing the given equation x² − 6x − 27 = 0 with this form gives α + β = 6 and αβ = −27. Rewrite the required expression by grouping the linear terms: αβ + 3α + 3β = αβ + 3(α + β). Substitution now gives −27 + 3(6) = −27 + 18 = −9. Therefore option A is correct. Option B is only the product αβ and ignores the remaining terms. Option C has the wrong sign, while option D results from an incorrect use of the root sum and product. No individual roots need to be calculated.
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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