Question 19280: Without drawing the graph of the given equation, determine (a) how many x-intercepts the parabola has and (b) whether its vertex lies above or below the x- axis.
Answer by mmm4444bot(95)   (Show Source):
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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
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