First slide

Theory of equations

Question

The quadratic $x^{2} + ax + b + 1 = 0$ has roots which are positive integers, then $(a^{2} + b^{2})$ canbe equal to

Moderate

A

1) 50
B

2) 37
C

3) 61
D

4) 19

Solution

$x^{2} + ax + b + 1 = 0,$ has positive integral roots $α and β$
Now, $(α + β) = - a and αβ = b + 1$
$\begin{array}{l} \Rightarrow (α + β)^{2} + (αβ - 1)^{2} = a^{2} + b^{2} \\ \Rightarrow a^{2} + b^{2} = (α^{2} + 1) (β^{2} + 1) \end{array}$
$\Rightarrow a^{2} + b^{2}$ can be equal to 50 (since other options have prime number)

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