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A quadratic equation is a type of polynomial equation of degree two. It typically takes the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and "\(a\)" should not be zero since that term gives the equation its quadratic nature. In this context:

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

To solve these equations, one commonly uses the quadratic formula, factoring, or completing the square, based on which suits the situation best. Solving a quadratic equation generally results in finding up to two solutions or roots for \(x\). These solutions are where the quadratic curve intersects the x-axis when plotted on a graph.
Understanding the basic structure of a quadratic equation is crucial because it allows you to apply these solutions. In our exercise, the equation is \(-x^2 + 16x - 62 = 0\), which is correctly identified with coefficients: \(a = -1\), \(b = 16\), and \(c = -62\). Knowing these helps us plug into the quadratic formula effectively.

The discriminant is a significant component in solving quadratic equations, as it provides valuable information regarding the nature and number of solutions. It is part of the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) and is denoted by \(D = b^2 - 4ac \).
Here's what the discriminant tells us:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there's exactly one real root, meaning the roots are equal.
• If \(D < 0\), the equation has no real roots, but two complex ones.

In our exercise, substituting the coefficients \(b = 16\), \(a = -1\), and \(c = -62\) gives us a discriminant of \(D = 8\). Since 8 is greater than zero, this indicates that our quadratic equation possesses two distinct real roots. Calculating the discriminant is a key step before moving on to find these roots, as it tells us what to expect in the solution process.

The roots of a quadratic equation are the solutions for \(x\) that satisfy the equation. Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we find the values of \(x\) where the quadratic expression equals zero. In simpler terms, they are the x-values where the graph of the quadratic intersects the x-axis.
After calculating the discriminant, the next step is to substitute it into the quadratic formula alongside the coefficients. For our specific example with a discriminant of 8:

• The equation inside the square root becomes \( \sqrt{8} \), which simplifies to \( 2\sqrt{2} \).
• The formula then becomes \( x = \frac{-16 \pm 2\sqrt{2}}{-2} \), yielding two solutions:
• \(x_1 = 8 - \sqrt{2}\)
• \(x_2 = 8 + \sqrt{2}\)

These reflect the roots of our quadratic equation. Breaking the steps down and understanding each part of the quadratic formula helps gain deeper insight into how the solutions arise and why they take their final form.