5) Sum of the roots of the quadratic equation is 5 and sum of their cubes is
35 then find the quadratic equation.
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Answer:
Let the two roots be α and β
α+β=5 , α
3
+β
3
=35
(α+β)
3
=α
3
+β
3
+3αβ(α+β)
(5)
3
=35+3αβ(5)
125=35+15αβ
90=15αβ
∴
15
90
=αβ
∴αβ=6
Quadratic Eq
n
:
x
2
−(sumofroots)x+productofroots=0
x
2
−(α+β)x+αβ=0
x
2
−5x+6=0
Step-by-step explanation:
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Solution:
According to the given condition,
But, a³+b³=(a+b)(a²+ab+b²)....{Formula}
So,
But,
Therefore,
Similarly, by splitting (2αβ)
We get,
But, α²+2αβ+β²= (α+β)²
...{(a+b)²=a²+2ab+b²}
...{Formula}
Therefore,
But, α+β=5
Therefore,
Therefore,
The equation will be..
...{By Formula}
Therefore, following is the required equation.
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