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The discriminant of a quadratic equation is a foundational concept that provides insight into the nature of its roots. The discriminant is represented by the symbol \(D\) and is calculated using the formula:

\[D = b^2 - 4ac\]
Here:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

The value of the discriminant helps in predicting the type of roots a quadratic equation has. Here's how:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, also referred to as a repeated or double root.
• If \(D < 0\), the equation has no real roots, but rather two complex roots.

In our particular problem, the discriminant value is zero, \(D = 0\), which indicates that there is exactly one real, repeated root.

When the discriminant is zero, the quadratic equation simplifies to a perfect square trinomial. This makes the square roots method an efficient and straightforward approach to finding the roots. Let's break down this method.

Given the quadratic equation:
\[h^2 + 18h + 81 = 0\]
It can be expressed as:
\[(h + 9)^2 = 0\]
This step involves recognizing that the original quadratic is a square of a binomial. By this recognition, we can directly solve the equation by taking the square root of both sides. Here's how:

• Taking the square root of both sides gives \(h + 9 = 0\).
• Solving for \(h\), we find \(h = -9\).

Because the square roots method led us to one solution, \(h = -9\), it confirms the discriminant’s indication of a double root. This method is efficient as it avoids the longer quadratic formula process, particularly when the quadratic neatly factors into a square of a binomial.

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, represented as:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula works universally for all quadratic equations and is especially helpful when factoring is difficult or impossible. It combines the coefficients \(a\), \(b\), and \(c\) directly into a calculation that reveals the roots.

Although versatile, using the quadratic formula to solve our example \(h^2 + 18h + 81 = 0\) is unnecessary given the discriminant is zero. For this exercise, because the equation developed into a perfect square, the square roots method provided a quicker solution.

However, more complex equations or those without perfect square trinomials benefit significantly from the quadratic formula. Remember, the formula always depends on calculating the discriminant, which determines the nature of the roots and guides us to the next steps or methods to employ.