find the value of k in the following exercise: kx^2-6x+1=0 if the equation has real, rational, and equal roots.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

For a quadratic equation ax^2 + bx + c = 0 to have real and equal roots b^2 = 4ac

Here we have kx^2 - 6x + 1 = 0

If the roots are real and equal

6^2 = 4*k*1

=> 36 = 4k

=> k = 9

The required value of k = 9

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

kx^2-6x+1=0

If the equation has real, rational, and equal roots then the discriminant has to be zero.

We'll calculate the discriminant delta:

delta = b^2 - 4ac

We'll identify a,b,c:

a = k, b = -6, c = 1

delta = 36 - 4k

We'll cancel delta:

36 - 4k = 0

We'll isolate k to the left side. For this reason, we'll subtract 36 both sides:

-4k = -36

We'll divide by -4:

k = 9

So, the roots of the equation are real and equal if and only if k = 9.

The roots are:

x1 = x2 =  (-b+sqrt delta)/2a

We'll substitute delta and b:

x1 = x2 =  (6 + 0)/2*9

x1 = x2 = 1/3

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