Intermediate Algebra | 34.5: The Discriminant and Parabola Intercepts
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Working through y = 2x² + 2x + 5, this lesson shows why a negative discriminant produces complex solutions — and what that means for the graph. You'll see how the vertex position above the x-axis and the sign of the discriminant together confirm a parabola with no x-intercepts. The video also addresses a common error in the vertex formula when the leading coefficient A is not equal to 1.
Key concepts covered:
• Vertex formula x = −B/(2A) — why A must appear in the denominator, not just 2
• Computing fractional vertex coordinates step by step: vertex at (−1/2, 9/2)
• How the leading coefficient A controls parabola width
• Applying the quadratic formula to test for x-intercepts
• The discriminant b² − 4ac and its three cases: positive (two intercepts), zero (one intercept), negative (no intercepts)
• Why a negative discriminant yields complex solutions rather than real ones
• Simplifying square roots of negative numbers: √(−36) = 6i, but √(−15) = i√15, not 15i
• Graphing a parabola with no x-intercepts using the vertex, y-intercept, and axis of symmetry