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68% (02:51) correct
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If α and β are the roots of the equation \(3x^2 – 6x + 4 = 0\), what is the value of \((\frac{α}{β} + \frac{β}{ α}) + 2(\frac{1}{ α} + \frac{1}{ β}) + 3αβ\)?
A. 2
B. 4
C. 6
D. 8
E. 16
A. 2
B. 4
C. 6
D. 8
E. 16
17 Feb 2023, 03:05
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\(3x^2–6x+4=0\)
α and β are the roots.
sum of the roots: α + β = -(-6/3) = 2... eqn 1
product of roots: αβ = 4/3....................eqn 2
we need to determine value of \((\frac{α}{β}+\frac{β}{α})+2(\frac{1}{α}+\frac{1}{β})+3αβ\) = ?
Divide eqn 1 by 2 we get
\(\frac{(α + β)}{αβ}\) = \((\frac{1}{α}+\frac{1}{β})\) = \(\frac{2}{(4/3)} \)= 3/2
square both sides of eqn 1
\(α^2 + β^2 +2αβ = 4\) ........eqn 3
divide eqn 3 by eqn 2
\( \frac{(α^2 + β^2 + 2αβ)}{αβ} = \frac{4}{(4/3)}\)
=> (α/β+β/α) + 2 = 3
=>\( (\frac{α}{β}+\frac{β}{α}) = 1\)
Now substitute the values of (α/β+β/α), (1/α+1/β) and αβ in the desired equation, we get
1 + 2*(3/2) +3*(4/3) = 8
Answer D
α and β are the roots.
sum of the roots: α + β = -(-6/3) = 2... eqn 1
product of roots: αβ = 4/3....................eqn 2
we need to determine value of \((\frac{α}{β}+\frac{β}{α})+2(\frac{1}{α}+\frac{1}{β})+3αβ\) = ?
Divide eqn 1 by 2 we get
\(\frac{(α + β)}{αβ}\) = \((\frac{1}{α}+\frac{1}{β})\) = \(\frac{2}{(4/3)} \)= 3/2
square both sides of eqn 1
\(α^2 + β^2 +2αβ = 4\) ........eqn 3
divide eqn 3 by eqn 2
\( \frac{(α^2 + β^2 + 2αβ)}{αβ} = \frac{4}{(4/3)}\)
=> (α/β+β/α) + 2 = 3
=>\( (\frac{α}{β}+\frac{β}{α}) = 1\)
Now substitute the values of (α/β+β/α), (1/α+1/β) and αβ in the desired equation, we get
1 + 2*(3/2) +3*(4/3) = 8
Answer D
26 Dec 2025, 06:11
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3x2–6x+4=0
a and b are the roots.
sum of the roots: a+b = -(-6/3) = 2
product of roots: a*b = 4/3
1. (a/b + b/a) = (a^2 + b^2)/ab = (a^2 +b^2 + 2ab -2ab) / ab = ((a+b)^2 -2ab)/ab = (2^2 - 2*4/3) / (4/3) = 1
2. 2*(1/a + 1/b) = 2*(a+b)/ab = 2*2/ (4/3) = 3
3. 3*ab = 3*4/3 = 4
1 + 3 + 4 = 8
Answer D
a and b are the roots.
sum of the roots: a+b = -(-6/3) = 2
product of roots: a*b = 4/3
1. (a/b + b/a) = (a^2 + b^2)/ab = (a^2 +b^2 + 2ab -2ab) / ab = ((a+b)^2 -2ab)/ab = (2^2 - 2*4/3) / (4/3) = 1
2. 2*(1/a + 1/b) = 2*(a+b)/ab = 2*2/ (4/3) = 3
3. 3*ab = 3*4/3 = 4
1 + 3 + 4 = 8
Answer D
