How do you know if a quadratic equation will have one, two, or no solutions?

It will always have two solutions: the question is are they real, complex, or a double root. Are you asking how does one know if the solutions are real?

One solution: When the lines intersect

No solutions: The same line
(There is no 2 solutions) It's infiently many solutions : when the lines are paralell

To determine how many solutions a quadratic equation will have, you can examine its discriminant. The discriminant is calculated using the formula: $\text{discriminant} = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the equation $ax^2 + bx + c = 0$.

1. If the discriminant is greater than zero ($\text{discriminant} > 0$), then the quadratic equation will have two distinct solutions. This means that the equation will intersect the x-axis at two different points.

2. If the discriminant is equal to zero ($\text{discriminant} = 0$), then the quadratic equation will have one real solution. In this case, the equation will intersect the x-axis at only one point (the vertex of the parabola).

3. If the discriminant is less than zero ($\text{discriminant} < 0$), then the quadratic equation will have no real solutions. The equation will not intersect the x-axis, and the solutions will involve complex numbers.

By evaluating the discriminant, you can determine the number and nature of solutions for a quadratic equation.