Theory:

Let us discuss how to determine the nature of the solutions for the given quadratic equations.
Example:
1. Find the nature of the solution of the equation \(y = x^2 - 2x - 3\).
 
Solution:
 
Step 1: Draw the graph of the equation \(y = x^2 - 2x - 3\).
 
The table of values for the equation \(y = x^2 - 2x - 3\) is:
 
\(x\)\(-2\)\(-1\)\(0\)\(1\)\(2\)\(3\)\(4\)
\(x^2\)\(4\)\(1\)\(0\)\(1\)\(4\)\(9\)\(16\)
\(2x\)\(-4\)\(-2\)\(0\)\(2\)\(4\)\(6\)\(8\)
\(3\)\(3\)\(3\)\(3\)\(3\)\(3\)\(3\)\(3\)
\(y\)\(5\)\(0\)\(-3\)\(-4\)\(-3\)\(0\)\(5\)
 
Step 2: Plot the points in the graph using a suitable scale.
 
4.png
 
Step 3: Join the points by a smooth curve.
 
Step 4: In the graph, observe that the curve intersects the \(X\) - axis at \(2\) points \((-1,0)\) and \((3,0)\). Therefore, the roots of the equation are \(-1\) and \(3\).
 
Since there are two points of intersection with \(X\) - axis, the given equation has real and unequal roots.
 
 
2. Find the nature of the solutions of the equation \(y = x^2 - 10x + 25\).
 
Solution:
 
Step 1: Draw the graph of the equation \(y = x^2 - 10x + 25\).
 
The table of values for the equation \(y = x^2 - 10x + 25\) is given by:
 
\(x\)\(3\)\(4\)\(5\)\(6\)\(7\)
\(x^2\)\(9\)\(16\)\(25\)\(36\)\(49\)
\(10x\)\(30\)\(40\)\(50\)\(60\)\(70\)
\(25\)
\(25\)
\(25\)
\(25\)
\(25\)
\(25\)
\(y\)\(4\)\(1\)\(0\)\(1\)\(4\)
 
Step 2: Plot the points in the graph.
 
5 (2).png
 
Step 3: Join the points by a smooth curve.
 
Step 4: Here, the curve meets the \(X\) - axis at only one point. Therefore, the point of intersection of the parabola with \(X\) - axis for the given equation is \((5,0)\).
 
Since the point of intersection is only one point with \(X\) - axis, the given quadratic equation has real and equal roots.