Solution (Quadratic Functions)
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
A solution of a quadratic equation is a value of x that satisfies the equation, meaning it makes the equation true. For a quadratic equation in the form:
ax^2 + bx + c = 0
The solutions are the values of x that, when substituted into the equation, result in 0 on the right-hand side.
Key points about solutions:
The solutions of a quadratic equation are also called the roots or x-intercepts because they represent the points where the graph of the quadratic function crosses the x-axis.
A quadratic equation can have:
Two real solutions: If the quadratic equation has two distinct values of x that make it equal to zero.
One real solution: If the quadratic equation has exactly one value of x that satisfies it (this occurs when the vertex of the parabola touches the x-axis).
No real solutions: If the quadratic equation has no real values of x that make it equal to zero because the parabola never crosses the x-axis
2 | 1 | 0 |
|---|---|---|
 |  |  |
Example:
For the quadratic equation x^2 -5x + 6 = 0, the solutions can be found by factoring:
(x - 2)(x - 3) = 0
So, the solutions are x = 2 and x = 3. These are the values of x that satisfy the equation.
In summary, the solutions of a quadratic equation are the values of x that make the equation true, and they represent the points where the graph of the quadratic crosses the x-axis.