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The discriminant is a key feature in quadratic equations used to determine the nature of their roots. It is calculated using the formula \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are coefficients from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). The discriminant tells us whether the solutions of the quadratic are real or complex:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, also known as a repeated or double root.
• If \( D < 0 \), the solutions are complex and appear in conjugate pairs.

In our example, the discriminant calculated is 529, a positive number, indicating that there are two distinct real solutions. This gives us an insight into the solution before we even start with actual calculations.

The quadratic formula is a reliable method for finding solutions to any quadratic equation. This formula is written as \( x = \frac{{-b \pm \sqrt{D}}}{{2a}} \), where \( D \) is the discriminant. The formula provides two solutions for \( x \), which are:

• The plus sign (\(+\)) in front of the square root gives the first solution.
• The minus sign (\(-\)) provides the second solution.

When applying the quadratic formula, it's essential to ensure the values of \( a \), \( b \), and \( D \) are substituted correctly. In the example exercise, \( a = 15 \), \( b = 17 \), and the discriminant \( D = 529 \). Substituting into the formula, you calculate:
For the first solution: \( x = \frac{{-17 + 23}}{{30}} = \frac{1}{5} \)
For the second solution: \( x = \frac{{-17 - 23}}{{30}} = -\frac{4}{3} \)
These solutions represent the points where the parabola described by the quadratic equation intersects the x-axis.

The nature of the solutions of a quadratic equation is dictated by its discriminant. In mathematics, understanding whether a quadratic will produce real or complex solutions is crucial for determining subsequent steps in problem-solving.
- **Real Solutions:** - Two distinct real solutions occur when the discriminant \( D > 0 \). This indicates that the graph of the quadratic crosses the x-axis at two different points. - One real solution happens when \( D = 0 \), meaning the graph touches the x-axis at a single point (vertex). This is called a vertex or repeated solution.- **Complex Solutions:** - When \( D < 0 \), the quadratic has two complex solutions. These do not intersect the x-axis at real number points instead, they form a complex conjugate pair. Complex solutions are important in scenarios involving oscillations and waves.In our specific problem, since \( D = 529 \) is larger than zero, we have two distinct real solutions: \( x = \frac{1}{5} \) and \( x = -\frac{4}{3} \), confirming the nature as predicted by the discriminant.