3. Examples to find the nature of the solutions

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:

 

 

Step 2: Plot the points in the graph using a suitable scale.

 

 

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 the \(X\) - 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:

 

 

Step 2: Plot the points in the graph.

 

 

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.