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Let `alphaandbeta` be two roots of quadratic equation `px^(2)=14x+8=0`, such that <br> `beta=6alpha` <br> `:."Sum of roots"=(-b)/(a)` <br> `impliesalpha+6alpha=-((-14))/(p)` <br> `implies7alpha-(14)/(p)impliesalpha=(2)/(p)` <br> Product of roots `-(c)/(a)` <br> `impliesalpha.6alpha=(8)/(p)` <br> `implies6alpha^(2)=(8)/(p)impliesalpha^(2)=(8)/(6p)` <br> From equations (1) and (2), we get <br> `((2)/(p))^(2)=(8)/(6p)implies(4)/(p^(2))=(4)/(3p)` <br> `impliesp^(2)=3p` <br> `impliesp^(2)-3p=0" "("don't cancel p both sides")` <br> `impliesp(p-3)=0` <br> :.Either p=0 or p=3. <br> But p=0 is not possible, as on putting, p=0 in the given equation, we don't have a quatratic equation and therefore we cannot get two roots. <br> Hence, p=3 <br> Alternatively, <br> Let one root of quadratic equation `px^(2)-14x+8=0` is `alpha`. <br> `:.palpha^(2)-14alpha+8=0" ".....(I)` <br> :. Other root of equation will be `6alpha`. <br> `:.p(6alpha^(2))-14(6alpha)+8=0` <br> `implies36palpha^(2)-84alpha+8=0` <br> `implies9palpha^(2)-21alpha+2=0" "......(2)` <br> Solving equations (1) and (2) by cross-multiplication method. <br> `:.(alpha^(2))/(-14(2)-8(-21))=(alpha)/(8(9p)-2p)=(1)/(p(-21)-9p(-14))` <br> `:.(alpha^(2))/(-28+168)=(alpha)/(70p)=(1)/(105p)` <br> `:.(alpha^(2))/(140)=(1)/(105p)and(alpha)/(70p)=(1)/(105p)` <br> `implies alpha^(2)=(140)/(105p)=(4)/(3p)andalpha=(70p)/(105p)=(2)/(3)` <br> `:.((2)/(3))^(2)=(4)/(3p)implies(4)/(9)=(4)/(3p)implies3p=9` <br> `:.p=3`**What is Quadratic Equation ?**

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