How come there are no x intercepts for x^2-x+(5/4)? Can you show me why, please?
To determine whether a quadratic equation like the one you provided, x^2 - x + (5/4), has x-intercepts or not, we need to look at the discriminant. The discriminant is the part of the quadratic formula that is inside the square root sign. In this case, the quadratic equation is in the form of ax^2 + bx + c, where a = 1, b = -1, and c = 5/4.
The discriminant (D) is given by the formula: D = b^2 - 4ac. In our case, substituting the values, we have: D = (-1)^2 - 4(1)(5/4).
Simplifying further, D = 1 - 5.
So, the discriminant D = -4.
The discriminant provides information about the nature of the roots of the quadratic equation:
1. If D > 0, there are two distinct real solutions (two x-intercepts).
2. If D = 0, there is one real solution (one x-intercept).
3. If D < 0, there are no real solutions (no x-intercepts).
Since the discriminant D = -4, which is less than zero, it means that the quadratic equation x^2 - x + (5/4) has no x-intercepts, or in other words, it does not intersect or cross the x-axis.
You can also visualize the quadratic equation on a graphing calculator or software to confirm that the graph does not intersect the x-axis.