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Question

For each equation given below, find the value (s) of p so that the equation has equal roots:

2px2-20x+13p-1=0


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Solution

Step 1: Solve for discriminant.

The given equation is 2px2-20x+13p-1=0.

Compare the given equation with the standard quadratic equation ax2+bx+c=0 to get:

a=2p,b=-20 and c=13p-1

Now, compute the discriminant as follows:

D=b2-4ac=-202-4×2p×13p-1=400-8p13p-1=400-104p2+8p

Step 2: Apply the condition for equal roots.

Since the roots are equal, therefore D=0.

b2-4ac=0400-104p2+8p=013p2-p-50=0

The quadratic formula used to solve the quadratic equation ax2+bx+c=0 is given as x=-b±b2-4ac2a.

Substitute a=13,b=-1 and c=-50 in the quadratic formula and simplify as follows:

p=--1±-12-4×13-502×13p=--1±512×13p=1±5126p=2,-2513

Thus, for p=2 or p=-2513, the equation has equal roots.


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