(i) Factorise 3397301 (you don’t need to factorise this into prime factors).
(ii) Solve $x^2 + 164x− 3 397 301 = 0$ without using the quadratic formula.
(i) $\lceil \sqrt{3397301} \rceil = 1844$, and $1844^2 - 3397301 = 3035$ which is not a perfect square.
$1845^2 - 3397301 = 82^2$, and so $3397301 = (1845-82)(1845+82) = 1927 \times 1763$.
The factorisation $3397301 = 1927 \times 1763$ is sufficient.
(ii) We notice that $1927-1763=164$, we have
$$ \begin{align} 0 & = x^2 + 164x - 3397301 \\ \\ & = x^2 +(1927-1763)x - (1927 \times 1763) \\ \\ &= (x+1927)(x-1763) \end{align} $$
And so the solutions are $x=-1927$ and $x+1763$.