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A quadratic equation is a polynomial equation of the second degree, generally written as \( ax^2 + bx + c = 0 \). In this standard form:\

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

Quadratic equations are very common in algebra and can describe a wide range of real-world phenomena. To solve these equations, you would typically use methods like factoring, completing the square, or the quadratic formula. The solutions to the quadratic equation, known as the roots, are crucial as they can determine the point where the quadratic curve intersects the x-axis. This understanding leads us to the concept of discriminants, which help determine the nature of these roots.

The discriminant is a vital component in determining the nature of the roots of a quadratic equation. For the standard quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is defined as \( D = b^2 - 4ac \). The value of \( D \) reveals important information about the roots of the quadratic equation:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), there is exactly one real root, also called a repeated or double root.
• If \( D < 0 \), the equation has no real roots, indicating that the solutions are complex numbers.

In the context of the function \( f(x) = \frac{x+3}{x^2 + 3cx + 6} \), we examined the denominator, \( x^2 + 3cx + 6 \). We needed this quadratic to have no real roots, thereby ensuring the function is defined for all real numbers \( x \). To achieve this, we set the discriminant \( 9c^2 - 24 \) to be less than zero.

Real numbers are the set of numbers that include both the rational and irrational numbers. They form the continuum of numbers you are most familiar with and can be found on the number line.

• Rational numbers include integers, fractions, and decimals that terminate or repeat.
• Irrational numbers are decimals that do not terminate or repeat (e.g., \( \pi \), the square root of 2).

A crucial concept with real numbers in the context of domains is their role in ensuring a function is defined. For the function \( f(x) = \frac{x+3}{x^2 + 3cx + 6} \), having a domain of all real numbers means the quadratic in the denominator should not equal zero for any real value of \( x \). By ensuring the discriminant is less than zero, \( 9c^2 - 24 < 0 \), it guarantees that the domain is all real numbers \( x \), as the denominator never becomes zero within this range of \( c \). Understanding real numbers is essential when dealing with functions because any restrictions on the domain (like excluding values that make a denominator zero) are based on real numbers.