Answer:
k = 3/5
Explanation:
You want the value of k that makes 5x² -2x +(2k-1) = 0 have equal roots.
Equal roots
The roots of the quadratic ax²+bx+c=0 will be equal when the discriminant d=b²-4ac is zero. Here, we have a=5, b=-2, c=(2k-1) and the discriminant is ...
d = b² -4ac = (-2)² -4(5)(2k-1)
d = 4 -20(2k-1) = 4 -40k +20 = 24 -40k
We want this to be zero, so ...
0 = 24 -40k
0 = 3/5 -k . . . . . . divide by 40, reduce the fraction
k = 3/5
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Additional comment
The attached graph shows the quadratic is tangent to the x-axis for k=3/5, meaning the roots are equal. They are x = 0.2.