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Answer:
5x^2 +10x = -5
Explanation:
The equation will have one solution when the discriminant is zero.
In standard form, the equation is ...
ax^2 +10x -c = 0
The discriminant of the equation ax^2+bx+c=0 is ...
b^2 -4ac
so the discriminant of your equation is ...
10^2 -4a(-c) = 100 +4ac
We want that to be zero, so we require ...
100 +4ac = 0
25 +ac = 0
ac = -25
Any pair of values of 'a' and 'c' that have a product of -25 will satisfy the requirement. As you state, one possible pair is a = 5 and c = -5. Putting these values into the given equation makes it ...
5x^2 +10x = -5 . . . . . the equation with a=5, c=-5
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Additional comment
Your problem statement does not require that 'a' be non-zero. If a=0, then the equation will have one real solution for any value of c.
If you restrict the solutions to integers, then possibilities include ...
(a, c) = (1, -25), (5, -5), (25, -1), (-25, 1), (-5, 5), (-1, 25)