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Solution of quadratic equations, Relation between roots and coefficients (only real roots to be considered). Practical problem on quadratic equations.

If the roots of the equation $x^2 - 10x + k - 1 = 0$ are α and β and $α^3 + β^3 = 70$,
then find the value of k.

Option:
A. 32
B. 30
C. 16
D. 70
Answer: A . 32

Justification:

Here, the roots of the equations $x^2 - 10x + k - 1 = 0$ are α and β:
Thus, α + β = ${-b}/a = (-10)/1 = 10$ and  $αβ = c/a = {k - 1}/1 = k - 1$
Now $α^3 + β^3 = 70$
.`. $(α + β)^3 - 3αβ(α + β) = 70$
.`. $(10)^3 - 3(k - 1)(10) = 70$
.`. 100 - 3k + 3 = 7
.`. 100 - 4 = 3k
.`. 96 = 3k
.`. k = $96/3$
.`. k = 32

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