# When solving an equation in the form ax^2=c by taking square root how many solutions will there be?

Jan 12, 2017

There can be $0$, $1$, $2$ or infinitely many.

#### Explanation:

Case $\boldsymbol{a = c = 0}$

If $a = c = 0$ then any value of $x$ will satisfy the equation, so there will be an infinite number of solutions.

$\textcolor{w h i t e}{}$
Case $\boldsymbol{a = 0 , c \ne 0}$

If $a = 0$ and $c \ne 0$ then the left hand side of the equation will always be $0$ and the right hand side non-zero. So there is no value of $x$ which will satisfy the equation.

$\textcolor{w h i t e}{}$
Case $\boldsymbol{a \ne 0 , c = 0}$

If $a \ne 0$ and $c = 0$ then there is one solution, namely $x = 0$.

$\textcolor{w h i t e}{}$
Case $\boldsymbol{a > 0 , c > 0}$ or $\boldsymbol{a < 0 , c < 0}$

If $a$ and $c$ are both non-zero and have the same sign, then there are two Real values of $x$ which satisfy the equation, namely $x = \pm \sqrt{\frac{c}{a}}$

$\textcolor{w h i t e}{}$
Case $\boldsymbol{a > 0 , c < 0}$ or $\boldsymbol{a < 0 , c > 0}$

If $a$ and $c$ are both non-zero but of opposite sign, then there are no Real values of $x$ which satisfy the equation. If you allow Complex solutions, then there are two solutions, namely $x = \pm i \sqrt{- \frac{c}{a}}$