Proving relations of quadratic equation with roots in some sequence" if a,b,c are in G.P., then equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common roof if da,cd,fc are in what sequence ( am, gm, hm etc) "" if α,β are roots of quadratic equation x2−x+p and γ,δ be the roots of equation x2−4x+q=0. If α−β,γ, and δ are in GP, then the integral values of p and q are respectively, are "How do I do questions like this?

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Proving relations of quadratic equation with roots in some sequence&quot; if a,b,c are in G.P., then equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common roof if da,cd,fc are in what sequence ( am, gm, hm etc) &quot;&quot; if α,β are roots of quadratic equation x2−x+p and γ,δ be the roots of equation x2−4x+q=0. If α−β,γ, and δ are in GP, then the integral values of p and q are respectively, are &quot;How do I do questions like this?

mistemePietsffi · Accepted Answer

Step 1Consider&amp;nbsp;ax2+2bx+c=0.As a, b, c are in GP, let&amp;nbsp;b=ar&amp;nbsp;and&amp;nbsp;c=ar2.So, we have,&amp;nbsp;ax2+2arx+ar2=0&amp;nbsp;i.e.&amp;nbsp;a(x+r)2=0&amp;nbsp;and hence&amp;nbsp;x=−r&amp;nbsp;is the only root of&amp;nbsp;ax2+2bx+c=0.This means,&amp;nbsp;x=−r&amp;nbsp;is also the root of&amp;nbsp;&amp;nbsp;dx&amp;nbsp;2+2ex+f=0&amp;nbsp;and we have&amp;nbsp;dr2−2er+f=0. Dividing this by&amp;nbsp;ar2&amp;nbsp;we get&amp;nbsp;da−2ear+far2=0da+fc=2ebHence,&amp;nbsp;da,;eb,;fc&amp;nbsp;are in AP.Step 2This same question is given in my textbook also and I think you&#039;ve got a typo in your question. Instead of&amp;nbsp;α−β&amp;nbsp;it should be&amp;nbsp;α,β. I&#039;ve solved the question accordingly.Let&amp;nbsp;(α,β,γ,δ)≡(a,ar,ar2,ar3).α+β=1⇒a(1+r)=1γ+δ=4⇒ar2(1+r)=4αβ=p⇒ar2=pγδ=q⇒a2r5=qNow, by first two equations we get(r,a)=(2,13),;(−2,−1)Hence, the integral value of&amp;nbsp;p,q&amp;nbsp;are&amp;nbsp;-2,&amp;nbsp;-32&amp;nbsp;respectively.

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