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When faced with any quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known coefficients, the quadratic formula is a reliable and straightforward method to find the unknown \(x\). The formula is given by \[x = \frac{-b \pm \sqrt{D}}{2a}\] where \(\sqrt{D}\) is the square root of the discriminant \(D\), and \(\pm\) symbolizes that there will be two solutions for \(x\).
To use this formula, you must have the values for \(a\), \(b\), and \(c\) from the equation. As seen in our example, we had \(a=2\), \(b=-4\), and \(c=-6\). Plugging these into our quadratic formula results in two potential values for \(x\), which are the roots of the equation. The beauty of the quadratic formula lies in its ability to provide solutions for any type of quadratic equation, whether the roots are real and distinct, real and equal, or complex.

The discriminant is a significant number computed from the coefficients of the quadratic equation, which tells us about the nature of the roots the equation possesses without solving it. It is represented by the symbol \(D\) and is calculated using the formula \[D = b^2 - 4ac\].
For the given equation \(2x^2 - 4x - 6 = 0\), we calculate the discriminant as follows:

• Identify \(a=2\), \(b=-4\), and \(c=-6\)
• Plug the values into the formula: \(D = (-4)^2 - 4(2)(-6)\)
• Simplify to find \(D = 16 + 48 = 64\)

The value of the discriminant (64 in this case) is positive, which tells us that the equation has two distinct real roots. If it were zero, the equation would have exactly one real root; and if it were negative, the roots of the equation would be complex.

The roots of a quadratic equation are the solutions to the equation; they are the values of \(x\) that make the quadratic equation true. Referring to our example, using the calculated discriminant \(D=64\) and the coefficients \(a=2\) and \(b=-4\), we can find the roots using the quadratic formula: \[x_1 = \frac{-(-4) + \sqrt{64}}{2\times2} = 3\] and \[x_2 = \frac{-(-4) - \sqrt{64}}{2\times2} = -1\].
This tells us that when \(x = 3\) or \(x = -1\), the original quadratic equation \(2x^{2} - 4x - 6 = 0\) will hold true. These roots can be real or complex and can vary from equation to equation, depending on the discriminant's value. Knowing the roots is essential as they are commonly sought in various mathematical problems, from geometry to calculus, and can provide insights into the characteristics of the equation's graph.