Determining the Values of ( k ) for Which the Equation ( (k-1)x^2 - k^2x^4 0 ) Has Two Equal Real Roots

In a quadratic equation of the form ( ax^2 bx c 0 ), the roots are equal if and only if the discriminant ( b^2 - 4ac ) is zero. This concept is fundamental in understanding when a quadratic equation has exactly one real root or two equal real roots. In this article, we will explore the specific condition on the parameter ( k ) for which the equation ( (k-1)x^2 - k^2x^4 0 ) results in two equal real roots. We will use the discriminant criterion to find the values of ( k ) that satisfy this condition.

Understanding the Discriminant

The discriminant of a quadratic equation ( ax^2 bx c 0 ) is given by ( b^2 - 4ac ). This discriminant tells us about the nature of the roots:

If ( b^2 - 4ac > 0 ), there are two distinct real roots. If ( b^2 - 4ac 0 ), there are two equal real roots. If ( b^2 - 4ac

Given Equation and Parameters

The given equation is ( (k-1)x^2 - k^2x^4 0 ). To apply the discriminant criterion, we need to rewrite this equation in standard quadratic form. First, we can factor out ( x^2 ) from the equation:

[ (k-1)x^2 - k^2x^4 0 ]

[ x^2[(k-1) - k^2x^2] 0 ]

This equation can be split into two parts:

( x^2 0 ) which gives ( x 0 ) ( (k-1) - k^2x^2 0 )

For the second part, we can solve for ( x ) by setting the quadratic equation to zero:

[ (k-1) - k^2x^2 0 ]

[ k^2x^2 k-1 ]

[ x^2 frac{k-1}{k^2} ]

[ x pm sqrt{frac{k-1}{k^2}} ]

Applying the Discriminant Criterion

For the equation ( (k-1)x^2 - k^2x^4 0 ) to have two equal real roots, we need to set the discriminant to zero:

[ b^2 - 4ac 0 ]

Here, the coefficients are:

( a k-1 ) ( b 0 ) ( c -k^2 )

Substituting these into the discriminant formula:

[ b^2 - 4ac 0^2 - 4(k-1)(-k^2) 0 ]

[ 4(k-1)k^2 0 ]

[ k^2(k-1) 0 ]

[ k 0 text{ or } k 1 ]

Verification of Solutions

Let's verify the values ( k 0 ) and ( k 1 ) by substituting them back into the original equation:

For ( k 0 ):

[ (0-1)x^2 - 0^2x^4 0 ]

[ -x^2 0 ]

There is only one real root ( x 0 ) (since ( -x^2 0 ) results in ( x 0 )).

For ( k 1 ):

[ (1-1)x^2 - 1^2x^4 0 ]

[ -x^4 0 ]

There is only one real root ( x 0 ) (since ( -x^4 0 ) results in ( x 0 )).

Conclusion

The equation ( (k-1)x^2 - k^2x^4 0 ) has two equal real roots when ( k 0 ) or ( k 1 ). These are the specific values of ( k ) for which the equation results in a double root.

In summary, the values of ( k ) that satisfy the condition for the equation ( (k-1)x^2 - k^2x^4 0 ) to have two equal real roots are ( k 0 ) and ( k 1 ).