When using the quadratic formula, why is only necessary to examine the discriminant to determine if the equation only has real number solutions?
The discriminant in the quadratic formula can be used to determine the nature of the solutions (real, complex, or repeated) of a quadratic equation. The discriminant is the expression under the square root in the quadratic formula, which is b^2 - 4ac.
If the discriminant is greater than zero (D > 0), it means that the quadratic equation has two distinct real solutions. In this case, the graph of the quadratic equation will intersect the x-axis at two different points.
If the discriminant is equal to zero (D = 0), it means that the quadratic equation has only one repeated real solution. In this case, the graph of the quadratic equation will only touch the x-axis at one point.
If the discriminant is less than zero (D < 0), it means that the quadratic equation has no real solutions. Instead, it will have two complex solutions. In this case, the graph of the quadratic equation will not intersect the x-axis at any point.
By examining only the discriminant, you can quickly determine whether the quadratic equation has real solutions or not. This is a useful shortcut because if the discriminant is less than zero, calculating the exact complex solutions requires additional steps, such as using the imaginary unit "i" to represent the square root of -1.