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Chapter 1: Problem 163

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

Short Answer

Expert verified
When a quadratic equation has imaginary solutions, its graph does not intersect the x-axis at any point.

Step by step solution

01

Understanding Imaginary Solutions

Firstly, we need to understand that imaginary solutions to a quadratic equation exist when the discriminant \(b^{2}-4 a c\) of the equation is less than zero.
02

Relating Solutions to Graph

The solutions of the quadratic equation corresponds to the x-values at the points where the quadratic graph intersects the x-axis. In other words, they correspond to the 'roots' of the function.
03

Implementing The Analysis to the Graph

When the solutions are imaginary, it means that the quadratic function does not intersect the x-axis at any point. This is because imaginary solutions exist only when the discriminant is less than zero, indicating no 'real' roots.

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