If (alpha) and (beta) are the roots of (2x^{2} – 5x + 6 = 0), find the equation whose roots are ((alpha + 1)) and ((beta + 1)).
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A.
(2x^{2} – 9x + 15 = 0) -
B.
(2x^{2} – 9x + 13 = 0) -
C.
(2x^{2} – 9x – 13 = 0) -
D.
(2x^{2} – 9x – 15 = 0)
Correct Answer: Option B
Explanation
Note: Given the sum of the roots and its product, we can get the equation using the formula:
(x^{2} – (alpha + beta)x + (alphabeta) = 0). This will be used later on in the course of our solution.
Given equation: (2x^{2} – 5x + 6 = 0; a = 2, b = -5, c = 6).
(alpha + beta = frac{-b}{a} = frac{-(-5)}{2} = frac{5}{2})
(alphabeta = frac{c}{a} = frac{6}{2} = 3)
Given the roots of the new equation as ((alpha + 1)) and ((beta + 1)), their sum and product will be
((alpha + 1) + (beta + 1) = alpha + beta + 2 = frac{5}{2} + 2 = frac{9}{2} = frac{-b}{a})
((alpha + 1)(beta + 1) = alphabeta + alpha + beta + 1 = 3 + frac{5}{2} + 1 = frac{13}{2} = frac{c}{a})
The new equation is given by: (x^{2} – (frac{-b}{a})x + (frac{c}{a}) = 0)
= (x^{2} – (frac{9}{2})x + frac{13}{2} = 2x^{2} – 9x + 13 = 0)