If the tangent to the curve $y=x^3-x^2+x$ at the point $(a,b)$ is also tangent to the curve $y=5x^2+2x-25$ at the point $(2,-1),$ then $3a+9b$ is equal to ___________.

$y=5x^2+2x-25$

$\therefore y\text{'}=10x+2\Rightarrow y\text{'}(x=2)=22$

So, the equation of tangent at $P(2,-1)$ is $y=22x-45\mathrm{............}\left(1\right)$

Now, $y=x^3-x^2+x$

$y^{\text{'}}=3x^2-2x+1$

${y^{\text{'}}}_{(x=a)}=3a^2-2a+1=\mathrm{22.............}\left(2\right)$

For common tangent Also $(a,b)$ lies on the curve

$y=x^3-x^2+x$

$\therefore b=a^3-a^2+a\mathrm{..........}\left(3\right)$

$\left(2\right)\Rightarrow a=3,\left(3\right)\Rightarrow b=21$

$[3a+9b]=198$