Question

Describe the relationship that must exist between a, b, and c in the equation ax^2+bx+c=0 in order for the equation to have exactly one solution

Answer 1

The equation
`ax^2+bx+c=0` has solutions given by
`x=(-b \pm √(b^2-4ac))/(2a)`. This means take the numbers a, b and c then plug them into that formula on your calculator and bam, answers.

The
`b^2-4ac` bit is called the discriminant and it tells you what kind of solutions occur,
-if
`b^2-4ac \ \textgreater \ 0` then the equation has two different solutions
-if
`b^2-4ac=0` then the equation has exactly one solution
-if
`b^2-4ac\ \textless \ 0` then the equation has no (real) solutions.

So you are interested in exactly one solution which happens if
`b^2-4ac=0` or equivalently
`b^2=4ac`.