= Solution
Cover the attached disc by a slightly enlarged interior and a collar of its boundary, and apply the <Seifert-van Kampen theorem>. The collar intersection has fundamental group generated by the boundary loop, whose image in $X$ is $[\alpha]$, while that loop becomes trivial in the disc. Van Kampen therefore quotients by the smallest normal subgroup containing $[\alpha]$:
$$
\boxed{\pi_1(X\cup_\alpha D^2,x_0)
\cong\pi_1(X,x_0)/\langle\!\langle[\alpha]\rangle\!\rangle}.
$$
This is the general <fundamental group after attaching a 2-cell>.
Start with a wedge of two circles, whose <fundamental group> is the free group
$$
\langle a,b\rangle.
$$
Attach a disc along the loop represented by $a^3b^{-7}$. The displayed quotient gives a connected two-dimensional <CW complex> $X$ with
$$
\boxed{\pi_1(X)\cong\langle a,b\mid a^3b^{-7}=1\rangle
=\langle a,b\mid a^3=b^7\rangle}.
$$
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