When solving for the roots of a quadratic equation, three (3) possibilities may arise what are these three possible cases? What do each of these cases mean?
http://en.wikipedia.org/wiki/Quadratic_equation
See the paragraphs under the heading Discriminant.
The three possible cases when solving for the roots of a quadratic equation are:
1. Two real and distinct roots: In this case, the quadratic equation has two different values of x that satisfy it. It means that the graph of the equation intersects the x-axis at two distinct points.
2. One real and repeated root: In this case, the quadratic equation has only one value of x that satisfies it. It means that the graph of the equation touches the x-axis at one point but does not cross it.
3. No real roots: In this case, the quadratic equation does not have any real values of x that satisfy it. It means that the graph of the equation does not intersect or touch the x-axis at any point.
These three cases depend on the discriminant (b^2 - 4ac) of the quadratic equation. If the discriminant is positive, there are two distinct real roots. If the discriminant is zero, there is one real repeated root. If the discriminant is negative, there are no real roots.
When solving a quadratic equation, we are looking for the values of x that make the equation equal to zero. The three possible cases that can arise when solving for the roots of a quadratic equation are:
1. Real and distinct roots: This means that the equation has two different values of x that make it equal to zero. Graphically, it represents a parabola that intersects the x-axis at two distinct points.
2. Real and equal roots: This case occurs when the equation has two identical values of x that make it equal to zero. Graphically, it represents a parabola that just touches the x-axis at one point and does not cross it.
3. Complex roots: This case happens when the equation has no real solutions. Instead, the solutions involve complex numbers. Complex roots occur when the discriminant of the quadratic equation (b^2 - 4ac) is negative. Graphically, it represents a parabola that does not intersect or touch the x-axis.
To determine which case applies to a quadratic equation, we need to compute its discriminant. The discriminant is the expression under the square root in the quadratic formula: b^2 - 4ac. By comparing the value of the discriminant to zero, we can determine the nature of the roots.