The function f(x), where f(x) = ax2 + bx + c, and a, b, c are constants, a ≠ 0, is called a quadratic function, or sometimes a quadratic polynomial. From elementary algebra $${(x + d)^2} \equiv {x^2} + 2dx + {d^2}.$$ Using this, we write $$a{x^2} + bx + c \equiv a\left( {{x^2} + \frac{b}{a}x + \frac{c}{a}} \right) \equiv a\left[ {{{\left( {x + \frac{b}{{2a}}} \right)}^2} + \frac{{4ac - {b^2}}}{{4{a^2}}}} \right] \Rightarrow a{x^2} + bx + c \equiv a{\left( {x + \frac{b}{{2a}}} \right)^2} + \frac{{4ac - {b^2}}}{{4a}}.$$ (3.1)