If the roots of the quadratic equation x² + px + q = 0 are tan 30° and tan 15°…
2019
If the roots of the quadratic equation x² + px + q = 0 are tan 30° and tan 15° respectively, then the value of 2 + q − p is:
- A.
1
- B.
3
- C.
0
- D.
2
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Correct answer: B
To find the value of 2 + q - p for the quadratic equation x^2 + px + q = 0 with roots tan(30) and tan(15), we use the relationship between roots and coefficients.
Step-by-Step Solution
Use Vieta's Formulas:
Sum of roots: tan(30) + tan(15) = -p
Product of roots: tan(30) * tan(15) = q
Calculate the Trigonometric Values:
tan(30) = 1 / sqrt(3)
tan(15) = tan(45 - 30) = (tan(45) - tan(30)) / (1 + tan(45) * tan(30))
tan(15) = (1 - 1 / sqrt(3)) / (1 + 1 / sqrt(3))
After rationalizing, tan(15) = 2 - sqrt(3)
Find p and q:
-p = (1 / sqrt(3)) + (2 - sqrt(3)) = (1 + 2 * sqrt(3) - 3) / sqrt(3) = (2 * sqrt(3) - 2) / sqrt(3) = 2 - (2 / sqrt(3))
So, p = (2 / sqrt(3)) - 2
q = (1 / sqrt(3)) * (2 - sqrt(3)) = (2 / sqrt(3)) - 1
Calculate 2 + q - p:
2 + ((2 / sqrt(3)) - 1) - ((2 / sqrt(3)) - 2)
2 + (2 / sqrt(3)) - 1 - (2 / sqrt(3)) + 2
2 - 1 + 2 = 3