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Real numbers are a set of numbers that include all rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot be neatly expressed as a simple fraction.
Real numbers also include whole numbers, integers, and decimal numbers. They are called "real" because they represent measurable quantities in our daily lives.

• Examples of Rational Numbers: Fractions like \( \frac{1}{2} \), whole numbers like 5, and terminating decimals like 0.75.
• Examples of Irrational Numbers: Non-terminating, non-repeating decimals such as \(\sqrt{2}\) or numbers like \( \pi \).

Real numbers are used in many kinds of calculations and are essential in solving equations.
A real number is what you typically think of as a "regular" number when not dealing with weird exceptions in math like imaginary numbers.

The Quadratic Formula is a handy tool used to find the roots of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \).
This formula provides an easy way to solve any quadratic equation.
The Quadratic Formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The key to using the Quadratic Formula is correctly identifying the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.

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

One important part of using this formula is considering the term \( b^2 - 4ac \).
The discriminant \(b^2 - 4ac\) determines the nature of the roots. When the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution.
However, if it is negative, as seen in the problem \( x^2 + 25 = 0 \), there are no real solutions.

Imaginary numbers are a unique set of numbers that arise when we try to take the square root of a negative number.
They are not part of the real number set, which means they are not encountered in everyday measurements or quantities.
Mathematicians introduced imaginary numbers to provide solutions to certain equations that do not have real solutions.

• Imaginary numbers are usually expressed in terms of \( i \), which represents \( \sqrt{-1} \).
• For example, \( \sqrt{-100} = 10i \).

Imaginary numbers are used with real numbers to form complex numbers.
This concept becomes especially useful in fields such as engineering and physics.
In the equation \( x^2 + 25 = 0 \), since the discriminant \( b^2 - 4ac = -100 \) is negative, we use imaginary numbers to find the solution, resulting in complex numbers as the roots.