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A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The quadratic equation is fundamental in algebra because its solutions are the points where the graph of the equation, a parabola, intersects the x-axis.
Quadratic equations can have:

• Two real and distinct solutions
• One real solution (called a double root)
• Two complex solutions

To solve these equations, the quadratic formula offers a reliable method. Using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] allows us to find the solutions by calculating the roots directly from the coefficients \(a\), \(b\), and \(c\). This formula considers all possible scenarios of root types, based on the value derived from the discriminant.

The discriminant of a quadratic equation gives insight into the nature of the roots of the equation without fully solving it. It is found using the expression \(b^2 - 4ac\), which is part of the quadratic formula.
Here's what the discriminant can tell us about the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root (or a double root), meaning the parabola touches the x-axis at a single point.
• If the discriminant is negative, the equation has two complex roots, and the parabola does not intersect the x-axis.

In the example given, the discriminant calculation \(5^2 - 4 \cdot 2 \cdot (-2)\) resulted in \(41\), which is a positive value, indicating two distinct real roots for the equation \(2x^2 + 5x - 2 = 0\).

The sum and product of the roots of a quadratic equation are closely tied to the equation's coefficients. These relationships provide additional methods to verify solutions.

• The sum of the roots \(x_1 + x_2\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by \( -\frac{b}{a} \).
• The product of the roots \(x_1 \cdot x_2\) is \( \frac{c}{a} \).

These expressions are derived from Vieta's formulas, which are consequences of the structure of quadratic equations.
In checking the equation given, the sum \(-\frac{5}{2}\) and the product \(-1\) calculated from the solutions matched the relationships derived from the coefficients, confirming the solutions were correctly found. This technique is especially useful as a check after using the quadratic formula.