A-7. Let a, b, c be real numbers wi...

A-7. Let a, b, c be real numbers with a 0 and let a, ß be the roots of the equation ax2 + bx + C = 0. Express the roots of a3x2 + abcx + c3 = 0 in terms of a, B akir hen

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( begin{array}{c}a x^{2}+b x+c=0 k+beta=-b a^{3} x^{2}+a b c x+c^{3}=0end{array} quad alpha beta=frac{c}{a} ) Lef rook of above equalion are ( 48 sqrt{5} ). [ begin{array}{l} y+f=-frac{a b c}{a^{3}}=-frac{b c}{a^{2}}=alpha beta(alpha+beta)-0 y delta=frac{c^{3}}{a^{3}}=alpha^{3} beta^{3} end{array} ] ( begin{aligned}(y-z)^{2} &=(y+p)^{2}+4 y z &=frac{b^{2}}{a}-frac{4 c^{3}}{a^{3}} leq frac{b^{2} c^{2}-4 a c^{3}}{a^{4}} end{aligned} ) ( begin{aligned} & eq frac{c^{2}}{a^{4}}left(b^{2} g-4 a cright) (y-delta)^{2} &=(4+r)^{2}-4 y v &=x^{2} beta^{2}(alpha+beta)^{2}-4 x^{3} p^{3} &=alpha^{2} beta^{2}left(-alpha^{2}+beta^{2}+2 x beta-4 alpha betaright) &=alpha^{2} beta^{2}(alpha-beta)^{2} end{aligned} ) ( Rightarrow quad y-delta=alpha beta(x-beta) ) Addiy ( 0,2,2 y=alpha beta(alpha+beta+alpha-beta)=2 alpha^{2} beta ) ( begin{aligned} & Rightarrow y=alpha^{2} beta Rightarrow quad &=frac{alpha^{3} beta^{3}}{gamma}=frac{alpha^{3} beta^{3}}{alpha^{2} beta}=alpha beta^{2} end{aligned} ) Then fore, ronts in terms of ( alpha ) f ( beta ) and ( alpha^{2} beta ) aud ( alpha beta^{2} )
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