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The quadratic formula is a powerful tool for solving quadratic equations, which are mathematical expressions of the form \( ax^2 + bx + c = 0 \). The general form of the quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). To use it, you need to identify the coefficients \(a\), \(b\), and \(c\) from your equation.

Let's apply this to our example \(3x^2 - 5x + 4 = 0\). Here, the coefficients are \(a = 3\), \(b = -5\), and \(c = 4\). Plugging these into the formula, we get \(x = \frac{5 \pm \sqrt{(-5)^2 - 4(3)(4)}}{6}\). This miraculous equation ensures you can systematically find the roots of any quadratic equation, even if they turn out to be complex numbers, which we'll discuss later.

The discriminant is a key part of solving quadratic equations and is denoted by the symbol \(\textbf{D}\). It is found using the formula \(\textbf{D} = b^2 - 4ac\) and tells us about the nature of the roots of the equation without actually solving it.

In our exercise, after identifying \(a = 3\), \(b = -5\), and \(c = 4\), we computed the discriminant as \(D = 25 - 48\) which equals \(D = -23\). A negative discriminant indicates that the equation has complex roots. When \(\textbf{D} > 0\), the quadratic has two distinct real roots. If \(\textbf{D} = 0\), it has exactly one real root (also known as a repeated or double root). Understanding the discriminant provides a clear path to determining the nature of the solutions even before you calculate them.

Complex numbers come into play when we encounter a negative discriminant in a quadratic equation. A complex number has a real part and an imaginary part. The imaginary unit, denoted by \(i\), is defined as the square root of \( -1 \).

Our example yielded roots in the form of complex numbers as the discriminant was \( -23 \). The quadratic formula then results in terms that include \( \sqrt{-23} \). Such square roots of negative numbers can't be simplified within the real number system and that's where \( i \) becomes useful. Here, the roots can be expressed as \( \frac{5}{6} \pm \frac{\sqrt{23} i}{6} \). Each solution has a real part, \( \frac{5}{6} \) and an imaginary part, \( \pm\frac{\sqrt{23}}{6} i \). Understanding complex numbers allows you to solve any quadratic equation, regardless of whether the discriminant is positive, zero, or negative.