The Complex Numbers. Trigonometric Sums. Infinite Products

Abstract

In order to solve the equation

$$a{x^2} + bx + c = 0,$$

((1.1))

where a, b, c are real numbers and a ≠ 0, for x ∈ ℝ, we use the identity

$$a{x^2} + bx + c = a\left[ {{{\left( {x + \frac{b}{{2a}}} \right)}^2} + \frac{{4ac - {b^2}}}{{4{a^2}}}} \right],$$

((1.2))

obtained by “completing the square.” A real number x satisfying (1.1) must satisfy

$${\left( {x + \frac{b}{{2a}}} \right)^2} = \frac{{{b^2} - 4ac}}{{4{a^2}}}.$$

((1.3))