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A quadratic equation is a specific type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This equation is characterized by having the highest power of \( x \) as 2. Because of this, the graph of a quadratic equation is a parabola, which can either open upwards or downwards depending on the sign of \( a \).

Quadratic equations can be solved using various methods including factoring, completing the square, and the quadratic formula. The quadratic formula is written as:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
This formula not only helps in finding the solutions of the quadratic equation but also involves the discriminant \( b^2 - 4ac \), which plays a significant role in determining the nature of the solutions.

The term "real solutions" in the context of quadratic equations refers to the root values of \( x \) that can be plotted on the real number line. These are the solutions where the quadratic equation intersects the x-axis on its graph.

To determine the number of real solutions of a quadratic equation, we utilize the discriminant \( b^2 - 4ac \). The discriminant gives insight into how many times, if any, the parabola will cross the x-axis:

• If the discriminant is greater than zero \((b^2 - 4ac > 0)\), there are two distinct real solutions.
• If it equals zero \((b^2 - 4ac = 0)\), there is one repeated real solution, meaning the parabola touches the x-axis at only one point.

In the case of the equation \( x^2 + 2x + 10 = 0 \), the discriminant is \(-36\), indicating it has no real solutions.

When a quadratic equation like \( x^2 + 2x + 10 = 0 \) has a discriminant less than zero \((b^2 - 4ac < 0)\), it implies the equation has complex roots. This means the solutions cannot be represented as real numbers, but rather as complex numbers, which have both a real and an imaginary component.

Complex roots always occur in conjugate pairs for quadratic equations, which means if \( x = a + bi \) is a solution, then \( x = a - bi \) is also a solution, where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit defined by \( i^2 = -1 \).

For the equation \( x^2 + 2x + 10 = 0 \), the discriminant \(-36\) confirms the presence of complex roots. Using the quadratic formula, one could find these specific roots, typically expressed in terms of the imaginary unit \( i \). This highlights the interplay between the discriminant and the nature of the quadratic equation's solutions.