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If a, b are the roots of x2 + px + q = 0, and w is a cube root of unity, then value of (ωα + ω2β)(ω2α + ωβ) is
- a)p2
- b)3q
- c)p2 – 2q
- d)p2 – 3q
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If a, b are the roots of x2 + px + q = 0, and w is a cube root of unit...
We have, α + β = -p,αβ = q
Now (ωα + ω2β)(ω2α + ωβ) = ω3α2 +ω4αβ+ω2αβ + ω3β2
= α2 +β2 +(ω+ω2)αβ = α2+β2 -αβ = (α + β)2 -3αβ = p2 -3q
Now (ωα + ω2β)(ω2α + ωβ) = ω3α2 +ω4αβ+ω2αβ + ω3β2
= α2 +β2 +(ω+ω2)αβ = α2+β2 -αβ = (α + β)2 -3αβ = p2 -3q
Most Upvoted Answer
If a, b are the roots of x2 + px + q = 0, and w is a cube root of unit...
Understanding the Problem
Given the quadratic equation \(x^2 + px + q = 0\), let \(a\) and \(b\) be its roots. By Vieta's formulas, we have:
- Sum of roots: \(a + b = -p\)
- Product of roots: \(ab = q\)
Let \(\omega\) be a primitive cube root of unity, satisfying \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).
Expression Breakdown
We need to evaluate the expression \((\omega a + \omega^2 b)(\omega^2 a + \omega b)\).
1. Expanding the Expression:
\[
(\omega a + \omega^2 b)(\omega^2 a + \omega b) = \omega^3 a^2 + \omega^2 ab + \omega a b + \omega^3 b^2
\]
Since \(\omega^3 = 1\), this simplifies to:
\[
a^2 + b^2 + (\omega^2 + \omega)ab = a^2 + b^2 - ab
\]
2. Calculating \(a^2 + b^2\):
Using the identity \(a^2 + b^2 = (a + b)^2 - 2ab\):
\[
a^2 + b^2 = (-p)^2 - 2q = p^2 - 2q
\]
3. Combining Results:
Substitute \(a^2 + b^2\) back into our expression:
\[
a^2 + b^2 - ab = (p^2 - 2q) - q = p^2 - 3q
\]
Final Result
Thus, the value of \((\omega a + \omega^2 b)(\omega^2 a + \omega b)\) is:
\[
\boxed{p^2 - 3q}
\]
This confirms that the correct option is \(d) p^2 - 3q\).
Given the quadratic equation \(x^2 + px + q = 0\), let \(a\) and \(b\) be its roots. By Vieta's formulas, we have:
- Sum of roots: \(a + b = -p\)
- Product of roots: \(ab = q\)
Let \(\omega\) be a primitive cube root of unity, satisfying \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).
Expression Breakdown
We need to evaluate the expression \((\omega a + \omega^2 b)(\omega^2 a + \omega b)\).
1. Expanding the Expression:
\[
(\omega a + \omega^2 b)(\omega^2 a + \omega b) = \omega^3 a^2 + \omega^2 ab + \omega a b + \omega^3 b^2
\]
Since \(\omega^3 = 1\), this simplifies to:
\[
a^2 + b^2 + (\omega^2 + \omega)ab = a^2 + b^2 - ab
\]
2. Calculating \(a^2 + b^2\):
Using the identity \(a^2 + b^2 = (a + b)^2 - 2ab\):
\[
a^2 + b^2 = (-p)^2 - 2q = p^2 - 2q
\]
3. Combining Results:
Substitute \(a^2 + b^2\) back into our expression:
\[
a^2 + b^2 - ab = (p^2 - 2q) - q = p^2 - 3q
\]
Final Result
Thus, the value of \((\omega a + \omega^2 b)(\omega^2 a + \omega b)\) is:
\[
\boxed{p^2 - 3q}
\]
This confirms that the correct option is \(d) p^2 - 3q\).
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If a, b are the roots of x2 + px + q = 0, and w is a cube root of unity, then value of (ωα+ω2β)(ω2α+ωβ) isa)p2b)3qc)p2 – 2qd)p2 – 3qCorrect answer is option 'D'. Can you explain this answer?
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