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Let α and β be the roots of the equation x2 + px + q = 0. If α3 and β3 are the roots of the equation x2 + mx + n = 0, then what is the value of m + n ?

Let α and β be the roots of the equation x2 + px + q = 0. If α3 and β3 are the roots of the equation x2 + mx + n = 0, then what is the value of m + n ? Correct Answer <span style="">p</span><span style=" line-height: 0; position: relative; vertical-align: baseline; top: -0.5em; font-size:10.5px;">3</span><span style=""> + q</span><span style=" line-height: 0; position: relative; vertical-align: baseline; top: -0.5em; font-size:10.5px;">3</span><span style=""> - 3pq</span>

Concept :

If α and β are the roots of quadratic equation ax2 + bx + c = 0

Sum of root α + β = -b/a

Product of root αβ = c/a

Formula used :

1. a3 + b3 = (a + b)(a2 + b2 - ab)

2. a2 + b2  = (a + b)2 - 2ab

Calculation :

Given that, α and β be the roots of the equation x2 + px + q = 0

⇒ α + β = -p        -----(1)

αβ = q                 -----(2)

Using the formula (1)

α3 + β3 = (α + β)(α2  + β2 - αβ) 

Again using the  formula (2)

α3 + β3 = (α + β)

α3β3 = n

⇒ n = q3         

⇒  m + n = p3 - 3pq + q3

∴  m +  n = p3 + q3 - 3pq

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