Question

. If a, ß are the roots of 7x^2
+ ax + 2 = 0 and if ß-a =
-13
Find the values of a.

​

Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:}}}}$

$\sf{The \ value \ of \ a \ is \ \frac{195+\sqrt105}{16} \ or \ \frac{195-\sqrt105}{16}}$

$\sf\orange{Given:}$

$\sf{The \ given \ quadratic \ equation \ is}$

$\sf{\implies{7x^{2}+ax+2=0}}$

$\sf{\implies{\beta-a=-13}}$

$\sf\pink{To \ find:}$

$\sf{The \ value \ of \ a.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{\implies{\beta-a=-13}}$

$\sf{\implies{\therefore{\beta=-13+a}}}$

$\sf{But \ \beta \ is \ the \ roots \ of \ given}$

$\sf{equation.}$

$\sf{\therefore{Substituting \ \beta \ in \ equation \ will \ give}}$

$\sf{result \ zero.}$

$\sf{\implies{\therefore{7(-13+a)^{2}+a(-13+a)+2=0}}}$

$\sf{\implies{7(169-26a+a^{2})-13a+a^{2}+2=0}}$

$\sf{\implies{1183-182a+7a^{2}-13a+a^{2}+2=0}}$

$\sf{\implies{7a^{2}+a^{2}-182a-13a+1183+2=0}}$

$\sf{\implies{8a^{2}-195a+1185=0}}$

$\sf{Here, \ a=8, \ b=-195 \ and \ c=1185}$

$\sf{By \ formula \ method}$

$\sf{b^{2}-4ac=(-195)^{2}-4(8)(1185)}$

$\sf{\implies{b^{2}-4ac=38025-37920}}$

$\sf{\implies{b^{2}-4ac=105}}$

$\sf{\implies{a=\frac{-b+\sqrt{b^{2}-4ac}}{2a} \ or \ \frac{-b-\sqrt{b^{2}-4ac}}{2a}}}$

$\sf{\implies{a=\frac{195+\sqrt105}{16} \ or \ \frac{195-\sqrt105}{16}}}$

$\sf\purple{\tt{\therefore{The \ value \ of \ a \ is \ \frac{195+\sqrt105}{16} \ or \ \frac{195-\sqrt105}{16}}}}$