Answer
a) $a$ and $c$ have opposite signs
b) $c=0$
c) $a$ and $c$ have the same sign
Work Step by Step
Assume $a\not=0$.
$$ax^2+c=0$$
$$x^2=-\frac{c}{a}$$
a) The equation has two solutions when $x^2>0$, so $\frac{c}{a}<0$, therefore when $a$ and $c$ have different signs.
b) The equation has one solution when $\frac{c}{a}=0$, therefore when $c=0$.
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c) The equation has no solution when $\frac{c}{a}>0$, therefore when $a$ and $c$ have the same sign.
