Puzzle: Finding Real Roots of an Equation

We all know to solve quadratic equation. We also know how to find the nature of roots of quadratic equation – whether it’s real and distinct/equal or imaginary.

Another way to solve a polynomial equation is to plot it. The point at which the equation crosses x-axis will be the solution. Try plotting a handful of equations with GeoGebra to familiarize yourself with the idea.

Puzzle: For what values of n does the equation $4x^2-(n-3)x^2+n=0$ have real roots?
Quora Question

One way is to use determinant of the quadratic equation to find the values of n.

The challenges for you is:

Solve the same problem by plotting it using GeoGebra or Desmos.
Find a third way to solve the same problem.

Purpose of this puzzle is to familiarize ourself with different ways to solve a polynomial equations, often not covered in depth in classrooms.

Leave your observations below in the comments! Those who plotted it using any software successfully, feel free to share the link to your file below.

If you’re tired, use the hints below:

Plotting to solve the question

Trick is in using a “slider” in GeoGebra. Create a slider n by simply giving “n” in input. Go to the new slider settings (using the 3 dots near slider), set lower and higher limits that you wish. Now input the equation you have in hand. Move the slider around to see when it cuts the real axis. Here’s the link to complete solution on GeoGebra.

Alternative solutions

Oops! Too many to list! Waiting for your observations in comments!