Problem 163 - Algebra and Trigonometry

Question

If a quadratic equation has imaginary solutions, how is this shown on the graph of \(y=a x^{2}+b x+c ?\)

Step-by-Step Solution

Answer

The graph of a quadratic equation with imaginary solutions does not intersect the x-axis.

Step 1: Understand Imaginary Solutions

Firstly, it's important to understand what imaginary solutions mean in a quadratic equation. Imaginary solutions occur when the discriminant \(b^{2}-4ac\) in the quadratic formula \(-\frac{b\pm \sqrt{b^{2}-4ac}}{2a}\) is negative, meaning there are no real solutions.

Step 2: Relation to Graph

On the graph of a quadratic equation, solutions are represented as points where the graph intersects with the x-axis. If the quadratic equation generates real solutions, the graph will intersect the x-axis at the points representing these solutions. However, having no real solutions (i.e., having imaginary solutions) effectively means that the graph of the quadratic equation does not intersect the x-axis at all.

Step 3: Conclusion

Hence, when a quadratic equation has imaginary solutions, it is graphically represented by the parabola not intersecting the x-axis at all.