To determine how many solutions a quadratic equation has, we use the discriminant, which is a part of the quadratic formula. The standard form of a quadratic equation is:
ax² + bx + c = 0
In this equation:
- a is the coefficient of x²
- b is the coefficient of x
- c is the constant term
The discriminant is calculated as:
D = b² – 4ac
Once you have the value of D, you can analyze it to determine the number of solutions:
- If D > 0, the quadratic equation has two distinct real solutions.
- If D = 0, there is one real solution (also called a repeated or double root).
- If D < 0, the equation has no real solutions, but it will have two complex solutions.
This method is useful because it allows you to quickly identify the nature of the solutions without having to solve the equation fully. Simply calculate the discriminant, and you’ll know how many solutions exist!