Question 19280
Hello There:

The x-intercepts of a quadratic equation are the solutions when y = 0.

-x^2 + 2*x - 1 = 0

The value of the discriminant that appears in the quadratic formula tells us how many solutions there are.

In case you've not memorized the quadratic formula, the discriminant is:

sqrt[b^2 - 4*(a)*(c)]

If the value of this expression is negative, then there are no solutions (thus, there are no x-intercepts).

If the value of this expression is zero, then there is one solution (thus, one x-intercept).

If the value of this expression is positive, then there are two solutions (thus, two x-intercepts).

In your equation we have a = -1, b = 2, and c = -1.  Therefore, the value of the discriminant is:

sqrt[(2)^2 - 4*(-1)*(-1)]

sqrt(4 - 4)

sqrt(0)

The value of the discriminant is zero, so there is one x-intercept.  Since there is only one x-intercept, it must be the vertex of the parabola that is touching the x-axis.  Therefore, the vertex does not lie above or below the x-axis.

~ Mark