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Finding the Intercepts of
To find the intercepts of the parabola :
Finding the y-intercept: Let and solve for :
When , . The y-intercept is the point .
Finding the x-intercepts: Let and solve for :
Before attempting to solve, compute the discriminant to predict how many solutions exist. Identify , , :
Because the discriminant is negative, the quadratic equation has no real solutions. Therefore, the parabola has no x-intercepts — it does not cross the x-axis.
This example demonstrates a case where using the discriminant saves time: rather than applying the full Quadratic Formula, the negative discriminant immediately reveals that no real x-values make , so the only intercept is the y-intercept .
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Ch.10 Quadratic Equations - Elementary Algebra @ OpenStax
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Learn After
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Intercept Analysis for a Facilities Maintenance Model
An operations analyst is using the equation to model the maintenance costs of a facility. To determine if the cost ever drops to zero, they check for the existence of x-intercepts using the discriminant method. Arrange the following steps in the correct order to perform this analysis as described in the course material.
An operations analyst is using the equation to model equipment wear and tear. To determine if the wear ever reaches zero (an x-intercept), the analyst calculates the discriminant using the formula . Based on the specific values in this equation, the numerical result of the discriminant calculation is ____.
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A quality control analyst uses the quadratic equation to model the variance in a manufacturing process. When checking for x-intercepts to determine if the variance ever reaches zero, which specific finding from the discriminant calculation () confirms that this model has no x-intercepts?
An operations analyst is reviewing the cost model . Match each mathematical component or calculation step with its correct numerical value based on the analysis of this specific equation.