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The discriminant is a powerful tool in solving quadratic equations. It helps us determine the nature of the roots without actually solving the equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated using the formula:\[D = b^2 - 4ac\]The value of the discriminant is crucial because:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root (or a repeated root).
• If \(D < 0\), there are two complex roots.

In our problem, the discriminant is \(-24\). Since it is negative, it implies that the quadratic equation has complex roots. This insight saves us time by indicating that we should expect solutions that involve imaginary numbers.

Complex roots arise when the discriminant of a quadratic equation is negative. This means the expression under the square root in the quadratic formula results in a negative number, which leads to imaginary numbers in the solution. Remember, the imaginary unit \(i\) is defined as \(\sqrt{-1}\). Therefore, when we calculate \(\sqrt{-24}\), it becomes \(2\sqrt{6}i\). In our equation, we find the complex roots using:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Substituting the values, the expression becomes:\[ x = \frac{-12 \pm 2\sqrt{6}i}{12} \]Thus, the solutions \(-1 + \frac{\sqrt{6}}{6}i\) and \(-1 - \frac{\sqrt{6}}{6}i\) are complex because they include \(i\), which indicates the presence of an imaginary component. Each solution consists of a real part \(-1\) and an imaginary part \(\pm \frac{\sqrt{6}}{6}i\).

The quadratic formula is an essential method for solving quadratic equations. It provides a straightforward way to find the roots by substituting values into a well-defined formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(b^2 - 4ac\) is known as the discriminant. This formula can solve any quadratic equation, provided \(a\), \(b\), and \(c\) are known.For the quadratic equation \(6x^2 + 12x + 7 = 0\), plugging in \(a = 6\), \(b = 12\), and \(c = 7\) into the quadratic formula gives:\[x = \frac{-12 \pm \sqrt{-24}}{12}\]Since \(\sqrt{-24}\) involves an imaginary number, the roots are complex.Follow these steps:

• Calculate \(\sqrt{-24} = 2\sqrt{6}i\).
• Substitute back to simplify further: \(x = \frac{-12 \pm 2\sqrt{6}i}{12}\).
• Separate into real and imaginary parts, yielding our complex solutions: \(-1 + \frac{\sqrt{6}}{6}i\) and \(-1 - \frac{\sqrt{6}}{6}i\).

The quadratic formula not only reveals the type of roots we should expect but also simplifies the process of solving any quadratic equation, ensuring we can accurately express solutions in terms of both real and imaginary components.