The roots of x^2-5x+6=0 are α and β. Which equation has roots α+1 and β+1?
Answer and explanation
Correct answer: x^2-7x+12=0
The original quadratic factors as (x-2)(x-3)=0, so α and β are 2 and 3 in either order. Adding 1 to each root gives the new roots 3 and 4. A monic quadratic with roots 3 and 4 is x^2-(3+4)x+(3)(4)=0, which simplifies to x^2-7x+12=0. Thus option A is correct. Using Vieta directly gives the same result: α+β=5 and αβ=6, so the new sum is α+β+2=7 and the new product is (α+1)(β+1)=αβ+α+β+1=6+5+1=12. Option B changes only the product, option C has the wrong transformed sum and product, and option D has the wrong sign.
Frequently asked questions
What is the correct answer to this question?
x^2-7x+12=0
Why is this the correct answer?
The original quadratic factors as (x-2)(x-3)=0, so α and β are 2 and 3 in either order. Adding 1 to each root gives the new roots 3 and 4. A monic quadratic with roots 3 and 4 is x^2-(3+4)x+(3)(4)=0, which simplifies to x^2-7x+12=0. Thus option A is correct. Using Vieta directly gives the same result: α+β=5 and αβ=6, so the new sum is α+β+2=7 and the new product is (α+1)(β+1)=αβ+α+β+1=6+5+1=12. Option B changes only the product, option C has the wrong transformed sum and product, and option D has the wrong sign.
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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