Solving by the Method of Perfect Squares

The method of perfect squares (also called "completing the square") solves $ax^2+bx+c=0$ by rewriting the quadratic expression so that the $x$-terms form a perfect-square trinomial, $(x+k)^2$, and then taking a square root on both sides.

Method:

1. Divide the whole equation by $a$ (the coefficient of $x^2$) so the leading coefficient becomes 1: $x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0$.
2. Move the constant term to the right side: $x^2 + \dfrac{b}{a}x = -\dfrac{c}{a}$.
3. Add $\left(\dfrac{b}{2a}\right)^2$ to both sides — this is exactly the number that makes the left side a perfect square.
4. The left side becomes $\left(x + \dfrac{b}{2a}\right)^2$. Take the square root of both sides (remembering the $\pm$ sign) and solve for $x$.

This method is important beyond being a solving technique in its own right — it is the derivation that produces Sridharacharya's formula in §5 (the syllabus asks only for the statement of that formula, but understanding where it comes from, via this exact perfect-square process, is what makes the formula meaningful rather than a number to memorize).