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In quadratic equations, the discriminant is a vital concept. It helps determine the nature of the roots of the equation. The discriminant, denoted as \(D\), is found using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation in the form \(ax^2 + bx + c = 0\).

Here's how the discriminant works:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, also known as a repeated root.
• If \(D < 0\), the roots are complex numbers, which means that they are not real.

In the original exercise, we calculated \(D = 360\) for \(x^2 - 18x - 9 = 0\). Because \(360 > 0\), this tells us that the equation has two distinct real roots. Always remember to calculate \(a\), \(b\), and \(c\) correctly before finding \(D\). Proper discrimination leads to a better understanding of the equation's behavior.

The quadratic formula is a universal method for finding the roots of any quadratic equation. It is especially useful when the equation doesn't factor neatly. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here's how it works:

• Start by identifying the coefficients \(a\), \(b\), and \(c\) from your quadratic equation in standard form.
• Use these coefficients to find the discriminant, \(b^2 - 4ac\). Once this step is complete, plug the values back into the formula.
• Solve for \(x\) to get the roots of the equation. The \(\pm\) symbol in the formula indicates that there will be two possible solutions: one adding the square root of the discriminant, and one subtracting it.

In the given exercise, with \(D = 360\), the formula becomes:\[x = \frac{18 \pm \sqrt{360}}{2}\]. This reveals two solutions: each performed carefully by handling \(\sqrt{360}\). Understanding the quadratic formula equips you with a reliable tool for solving diverse quadratic equations efficiently.

The standard form of a quadratic equation is crucial for analysis and solving. It is written as \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are known coefficients.
• \(a\), the coefficient of \(x^2\), must not be zero.
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

Rearranging an equation into standard form allows you to easily identify these coefficients, which is essential in applying further methods like using the discriminant or quadratic formula.

In the initial step of the exercise, the equation \(x^2 - 18x = 9\) was rearranged to become \(x^2 - 18x - 9 = 0\). This rearrangement aligns it with the standard form, making it straightforward to identify \(a = 1\), \(b = -18\), and \(c = -9\). Understanding how to convert to this form is foundational, providing a clear path forward to solve any quadratic equation.