A quadratic equation is a second-degree polynomial equation, which means it contains at least one squared term. It can be written in the general form:
\[ ax^2 + bx + c = 0 \]
where:
- \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero,
- \( x \) is the variable representing the unknown,
- \( ax^2 \) is the quadratic term,
- \( bx \) is the linear term,
- \( c \) is the constant term.
Quadratic equations can have one of three different types of solutions, depending on the discriminant \( b^2 - 4ac \):
1. **Two Real Solutions**: If the discriminant is positive (\( b^2 - 4ac > 0 \)), then the equation has two distinct real roots.
2. **One Real Solution**: If the discriminant is zero (\( b^2 - 4ac = 0 \)), then the equation has one real root, also known as a repeated root or a double root.
3. **No Real Solutions**: If the discriminant is negative (\( b^2 - 4ac < 0 \)), then the equation has no real roots. In this case, the solutions are complex conjugates.
Quadratic equations can be solved using various methods, including factoring, completing the square, and using the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
This formula gives the values of \( x \) where the quadratic equation equals zero. Quadratic equations are widely used in mathematics, physics, engineering, economics, and other fields to model various phenomena, such as projectile motion, population growth, and financial investments.