Find the value of k = ( x^2 - 4 )/ ( 2x -5 ) if the roots of the equation are equal .
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We have k = (x^2 - 4)/( 2x - 5)
k = (x^2 - 4)/( 2x - 5)
=> x^2 - 4 = 2kx - 5k
=> x^2 - 2kx + 5k - 4 = 0
As the roots of the quadratic equation are equal, b^2 - 4ac = 0
=> (-2k)^2 - 4*( 5k - 4) = 0
=> 4k^2 - 20k + 16 = 0
=> k^2 - 5k + 4 = 0
=> k^2 - 4k - k + 4 = 0
=> k(k - 4) - 1(k - 4) = 0
=> (k-1)(k-4) = 0
=> k = 1 and k = 4
Therefore the values of k are 1 and 4.
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We'll multiply both sides by 2x - 5.
k(2x - 5) = (x^2 - 4)(2x - 5)/ (2x - 5)
We'll simplify and we'll get:
k(2x - 5) = (x^2 - 4)
We'll remove the brackets:
2kx - 5k = x^2 - 4
We'll move all terms to one side:
x^2 - 4 - 2kx + 5k = 0
We'll combine like terms:
x^2 - 2kx + 5k - 4 = 0
For the roots of the quadratic to be equal, the discriminant delta has to be zero.
delta = b^2 - 4ac
a,b,c are the coefficients of the quadratic.
delta = (-2k)^2 - 4(5k - 4)
delta = 4k^2 - 20k + 16
4k^2 - 20k + 16 = 0
We'll divide by 4:
k^2 - 5k + 4 = 0
We'll apply quadratic formula:
k1 = [5 + sqrt(25 - 16)]/2
k1 = (5+3)/2
k1 = 4
k2 = (5-3)/2
k2 = 1
The values of k, for the equation to have equal roots, are: {1 ; 4}.
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