Solve for x, x^2-(2 b-1) x+ (b^2-b-20 )=0 - Vidyadip

Question

Solve for $x, x^2-(2 b-1) x+\left(b^2-b-20\right)=0$

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Answer

Here, the given equation is
$x^2-(2 b-1) x+\left(b^2-b-20\right)=0$
Finding the diseriminant
$D=b^2-4 a c$
$D=(-(2 b-1))^2-4 \times 1 \times\left(b^2-b-20\right)$
$D=4 b^2+1-4 b-4\left(b^2-b-20\right)$
${\left[\text { using }(a-b)^2=a^2+b^2-2 a b\right]}$
$D=4 b^2+1-4 b-4 b^2+4 b+80$
$D=81$
Now,
$x=\frac{-b \pm \sqrt{D}}{2 a}$
$x=\frac{-(-(2 b-1) \pm \sqrt{81}}{2}$
$x=\frac{2 b-1 \pm 9}{2}$
$x=\frac{2 b-1+9}{2}, \frac{2 b-1-9}{2}$
$x=\frac{2 b+8}{2}, \frac{2 b-10}{2}$
$x=b+4, b-5$

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