a_3 x^3 −x^2 +a_1 x−7=0 is a cubic polynomial in x whose Roots are α , β , γ positive real numbers satisfying ((225α^2 )/(α^2 +7))=((144β^2 )/(β^2 +7))=((100γ^2 )/(γ^2 +7)) find (a


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Question Number 195027 by York12 last updated on 22/Jul/23

$a_{3} x^{3} - x^{2} + a_{1} x - 7 = 0 i s a c u b i c p o l y n o m i a l i n x$ $w h o s e R o o t s a r e α, β, γ p o s i t i v e r e a l n u m b e r s$ $s a t i s f y i n g$ $\frac{225 α^{2}}{α^{2} + 7} = \frac{144 β^{2}}{β^{2} + 7} = \frac{100 γ^{2}}{γ^{2} + 7}$ $f i n d (a_{1})$
Commented by mr W last updated on 25/Jul/23

$i g o t a_{1} = \frac{77}{15}$
Commented by York12 last updated on 25/Jul/23

$p l e a s e s i r e x p l a i n w h y$
	Answered by mr W last updated on 25/Jul/23

	$\Rightarrow α + β + γ = \frac{1}{a_{3}}$ ${(\frac{1}{x})}^{3} - \frac{a_{1}}{7} {(\frac{1}{x})}^{2} + \frac{1}{7} (\frac{1}{x}) - \frac{a_{3}}{7} = 0$ $\Rightarrow \frac{a_{1}}{7} = \frac{1}{α} + \frac{1}{β} + \frac{1}{γ}$ $a_{3} x^{3} - x^{2} + a_{1} x - 7 = 0$ $x^{2} + 7 = x (a_{3} x^{2} + a_{1})$ $\frac{225 α^{2}}{α^{2} + 7} = \frac{144 β^{2}}{β^{2} + 7} = \frac{100 γ^{2}}{γ^{2} + 7} = \frac{1}{k}, s a y$ $\frac{225 α^{2}}{α (a_{3} α^{2} + a_{1})} = \frac{144 β^{2}}{β (a_{3} β^{2} + a_{1})} = \frac{100 γ^{2}}{γ (a_{3} γ^{2} + a_{1})} = \frac{1}{k}$ $\Rightarrow a_{3} α + \frac{a_{1}}{α} = 225 k$ $\Rightarrow a_{3} β + \frac{a_{1}}{β} = 144 k$ $\Rightarrow a_{3} γ + \frac{a_{1}}{γ} = 100 k$ $\Rightarrow a_{3} (α + β + γ) + a_{1} (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = 469 k$ $\Rightarrow 1 + \frac{a_{1}^{2}}{7} = 469 k$ $\Rightarrow a_{1} = \sqrt{7 (469 k - 1)}$ $\frac{225}{1 + \frac{7}{α^{2}}} = \frac{144}{1 + \frac{7}{β^{2}}} = \frac{100}{1 + \frac{7}{γ^{2}}} = \frac{1}{k}$ $\Rightarrow \frac{\sqrt{7}}{α} = \sqrt{225 k - 1}$ $\Rightarrow \frac{\sqrt{7}}{β} = \sqrt{144 k - 1}$ $\Rightarrow \frac{\sqrt{7}}{γ} = \sqrt{100 k - 1}$ $\sqrt{7} (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}$ $\frac{a_{1}}{\sqrt{7}} = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}$ $\Rightarrow \sqrt{469 k - 1} = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}$ $\Rightarrow \sqrt{225 k - 1} + \sqrt{144 k - 1} = \sqrt{469 k - 1} - \sqrt{100 k - 1}$ $\Rightarrow \sqrt{(225 k - 1) (144 k - 1)} = 100 k - \sqrt{(469 k - 1) (100 k - 1)}$ $\Rightarrow 245 k - 2 = 2 \sqrt{(469 k - 1) (100 k - 1)}$ $\Rightarrow 127575 k = 1296$ $\Rightarrow k = \frac{1296}{127575} = \frac{16}{1575}$ $a_{1} = \sqrt{7 (\frac{469 \times 16}{1575} - 1)} = \frac{77}{15} ✓$ $α = \sqrt{\frac{7}{225 \times \frac{16}{1575} - 1}} = \frac{7}{3}$ $β = \sqrt{\frac{7}{144 \times \frac{16}{1575} - 1}} = \frac{35}{9}$ $γ = \sqrt{\frac{7}{100 \times \frac{16}{1575} - 1}} = 21$ $a_{3} = \frac{1}{\frac{7}{3} + \frac{35}{9} + 21} = \frac{9}{245}$
	Commented by York12 last updated on 25/Jul/23

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