Note that, since we are told that a≠0, the right-hand side of the obtained equation is not undefined. We will now see what must be true of a and c in each of the cases.
The Equation Has Two Solutions
If the equation ax^2+c=0 has two solutions, it means the equivalent equation we obtained above has two solutions.
x^2=- c/a ⇔ x=± sqrt(- c/a)
Therefore, not only is sqrt(- ca) a real number but the positive and negative values created by the plus or minus symbol must be different.
The Square Root Must Be Real
First, let's think about what must be true in order for the square root to be a real number. For sqrt(- ca) to be real, the value under the square must be non-negative. This, in turn, implies that ca is less than or equal to 0.
- c/a≥ 0 ⇔ c/a≤ 0
For this to happen, it must be true that either c=0 or a and c have opposite signs.
The Values Must Be Different
Let's consider what it means that the values must be different.
sqrt(- c/a) ≠- sqrt(- c/a)
Because of the negative, these expressions would only ever be equal if both sides were 0. This means that we need to exclude any value(s) that would cause these expressions to equal 0.
These expressions will only be equal when c=0. Therefore, in order to have two unique solutions to the quadratic equation, we need c≠0.
Conclusion
We can conclude that if ax^2+c=0 has two solutions and a≠0, then a and c must have opposite signs — one positive and one negative — and c cannot be 0.
The Equation Has Only One Solution
If the equation ax^2+c=0 has only one solution, it means the equivalent equation we obtained above also has only one solution.
x^2=- c/a ⇔ x=± sqrt(- c/a)
Therefore, not only is sqrt(- ca) a real number but also the positive and negative values created by the plus or minus symbol must be equal. As discussed above, we know that the square root is real when a is not equal to 0 and the expressions are equal when c=0.
sqrt(- c/a) = - sqrt(- c/a) ⇒ la≠0 c=0
If ax^2+c=0 has only one solution and a≠0, then c=0.
The Equation Has No Solutions
If the equation ax^2+c=0 has no solutions, it means the equivalent equation we obtained above has no solutions.
x^2=- c/a ⇔ x=± sqrt(- c/a)
Therefore, we need to know when sqrt(- ca) is not real. For this to be the case, the value under the square root must be negative. This, in turn implies that ca is greater than 0.
- c/a< 0 ⇔ c/a> 0
The fraction ca will be positive only if both a and c have the same sign. In conclusion, assuming a≠0, if ax^2+c=0 has no solutions, then a and c have the same sign — both positive or both negative — and c is not equal to 0.
