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The quadratic formula is a powerful tool for solving quadratic equations. It provides an efficient way to find the solutions to any equation of the form \(ax^2 + bx + c = 0\). The formula itself is:

• \(\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The quadratic formula works by calculating the roots of the quadratic equation based on the coefficients \(a\), \(b\), and \(c\). These coefficients come from the standard form of the quadratic equation. By substituting these values into the formula, we can determine the solution set for \(x\).
This method is especially useful when factoring is difficult or impossible, or when the equation itself does not seem to fit a recognizable pattern.

When solving quadratic equations, the term under the square root in the quadratic formula, known as the discriminant, can be negative. A negative discriminant indicates that the equation has complex solutions rather than real ones. In mathematical terms, these are solutions that involve the imaginary unit \(i\), where \(i = \sqrt{-1}\).
For example, when calculating \(\sqrt{-12}\), you can rewrite it as \(i\sqrt{12}\) because \(\sqrt{-1} = i\). Then further simplify \(\sqrt{12}\) to \(2\sqrt{3}\).
Consequently, having a negative discriminant, the solutions are not just numerical values but involve complex numbers, expressed as \(a + bi\) and \(a - bi\), showing both a real part \(a\) and an imaginary component \(bi\). This happens because the discriminant affects the type of solution you get.

The discriminant is a pivotal value in the quadratic formula that greatly influences the type of solutions you'll find. It's calculated as \(b^2 - 4ac\) and can inform us of several conditions based on its value:

• If the discriminant is positive, expect two distinct real solutions.
• If the discriminant is zero, expect exactly one real solution (a repeated root).
• If the discriminant is negative, the solutions will be complex or imaginary.

In our problem, the discriminant is \(-12\), calculated using \(6^2 - 4 \times 1 \times 12 = -12\). This negative value confirms that the quadratic equation has complex solutions. Understanding how the discriminant works can predict the nature of the roots before performing detailed calculations, providing insight into what type of solutions to expect.