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if α, β are the roots of the equation ax² + bx + c = 0, then the value of α³ + β³ is
- a)[(3abc + b³)/a³]
- b)[(a³ + b³)/3ab]
- c)[(3abc - b³)/a³]
- d)[(b³ - 3abc)/a³]
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
if α, β are the roots of the equation ax² + bx + c = 0...
(α+β)3=α3+3α2β+3αβ2+β3
or we can make
α3+β3=(α+β)3−3α2β+3αβ2
α3+β3=(α+β)3−3αβ(α+β)
and :
(α+β)=−ba
αβ=ca
finally we got
x2−(α3+β3)x+(αβ)3=x2−((α+β)3−3αβ(α+β))x+(αβ)3
the we enter the values :
x2−((−ba)3−3(ca−ba))+(ca)3
and final result is :
a3x2−(−b3+3abc)x+c3
or
a3x2+(b3−3abc)x+c3
Most Upvoted Answer
if α, β are the roots of the equation ax² + bx + c = 0...
Understanding the Problem
Given the quadratic equation ax² + bx + c = 0, we know that its roots α and β can be expressed using Vieta's formulas:
- Sum of roots: α + β = -b/a
- Product of roots: αβ = c/a
We need to find the value of α³ + β³.
Using the Identity for Sum of Cubes
The sum of cubes can be expressed as:
α³ + β³ = (α + β)(α² - αβ + β²)
Now we can substitute α + β and αβ from Vieta's formulas:
Calculating α² + β²
To find α² + β², we can use the identity:
α² + β² = (α + β)² - 2αβ
Substituting the values:
α² + β² = (-b/a)² - 2(c/a)
Now, we can rewrite α³ + β³:
Final Calculation
Substituting everything back into the equation:
α³ + β³ = (-b/a)((-b/a)² - 2(c/a)) + 3c/a
Simplifying this gives:
Final Expression
After simplification, we arrive at:
α³ + β³ = (3abc - b³)/a³
Thus, the correct option is (c) [(3abc - b³)/a³].
This confirms why option 'C' is the right choice.
Given the quadratic equation ax² + bx + c = 0, we know that its roots α and β can be expressed using Vieta's formulas:
- Sum of roots: α + β = -b/a
- Product of roots: αβ = c/a
We need to find the value of α³ + β³.
Using the Identity for Sum of Cubes
The sum of cubes can be expressed as:
α³ + β³ = (α + β)(α² - αβ + β²)
Now we can substitute α + β and αβ from Vieta's formulas:
Calculating α² + β²
To find α² + β², we can use the identity:
α² + β² = (α + β)² - 2αβ
Substituting the values:
α² + β² = (-b/a)² - 2(c/a)
Now, we can rewrite α³ + β³:
Final Calculation
Substituting everything back into the equation:
α³ + β³ = (-b/a)((-b/a)² - 2(c/a)) + 3c/a
Simplifying this gives:
Final Expression
After simplification, we arrive at:
α³ + β³ = (3abc - b³)/a³
Thus, the correct option is (c) [(3abc - b³)/a³].
This confirms why option 'C' is the right choice.
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