1. Tardigrade
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  4. Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x2 + 2px + q = 0 and α, (1/β) are the roots of the equation x2 + 2bx + c = 0, where β2 ∉ (-1, 0, 1) Statement-1:(p2 - q)(b2 - ac) ge 0 Statement-2: b ≠ pa or c ≠ qa

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Q. Let a, b, c, p, q be real numbers. Suppose $\alpha, \beta$ are the roots of the equation $x^2 + 2px + q = 0$ and $\alpha, \frac{1}{\beta}$ are the roots of the equation $x^2 + 2bx + c = 0$, where $\beta^{2} \notin \left(-1, 0, 1\right)$
Statement-1:$\left(p^{2} - q\right)\left(b^{2} - ac\right) \ge 0$
Statement-2: $b \ne pa$ or $c \ne qa$

Complex Numbers and Quadratic Equations

Solution:

Given, $x^2 + 2px + q = 0$
$\therefore \quad \alpha+\beta = -2p\quad\dots\left(i\right)$
$\alpha\beta = q\quad \dots \left(ii\right)$
And $ax^{2} + 2bx + c = 0$
$\therefore \quad \alpha +\frac{1}{\beta} = - \frac{2b}{a}\quad \dots \left(iii\right)$
and $\frac{\alpha}{\beta} = \frac{c}{a}\quad \dots \left(iv\right)$
Now, $\left(p^{2} - q\right)\left(b^{2} - ac\right)$
$= \left[\left(\frac{\alpha +\beta }{-2}\right)^{2} -\alpha\beta\right]\left[\left(\frac{ \alpha +\frac{1}{\beta }}{2}\right)^{2}- \frac{\alpha}{\beta}\right]a^{2}$
$= \frac{\left(\alpha -\beta\right)^{2} }{16} \left(\alpha -\frac{1}{\beta }\right)^{2} . a^{2} \ge 0$
$\therefore \quad$ Statement 1 is true.
Again, now $pa = - \left(\frac{\alpha +\beta }{2}\right)a = -\frac{a}{2}\left( \alpha +\beta \right)$
and $b = -\frac{a}{2}\left(\alpha +\frac{1}{\beta }\right)$
Since, $pa \ne b\quad\Rightarrow \quad\alpha +\frac{1}{\beta } \ne \alpha +\beta $
$\Rightarrow \quad \beta^{2} \ne 1, \beta \ne \left\{-1, 0, 1\right\}$, which is correct,
Similarly, if $c \ne qa\quad\Rightarrow \quad a \frac{\alpha }{\beta } \ne a \alpha\beta$
$\Rightarrow \quad\alpha \left(\beta -\frac{1}{\beta}\right)\ne0$
$\Rightarrow \quad\alpha \ne 0$ and $\beta -\frac{1}{\beta }\ne 0$
$\Rightarrow \quad\beta \ne\left\{-1, 0, 1\right\}$
Statement 2 is true.
Both Statement 1 and Statement 2 are true. But
Statement 2 does not explain statement 1.