# Thread: imaginary numbers x^2 + a^2 = 0 question

1. ## imaginary numbers x^2 + a^2 = 0 question

my edexcel further maths book says i need to solve this equation giving the answer in terms of the positive number a.
x^2 + a^2 = 0.

the answer in the back of the text book is x = + or - ai
but i dont see how i can get this answer

2. ## Re: imaginary numbers x^2 + a^2 = 0 question

Whenever you have the square root of a negative value, you may change the sign under the radical and place an i outside the radical as a factor in its place.

Another way to look at it is:

$\displaystyle x^2+a^2=0$

$\displaystyle x^2-(-1)a^2=0$

Since $\displaystyle i^2=-1$ we may write

$\displaystyle x^2-i^2a^2=0$

$\displaystyle x^2-(ai)^2=0$

$\displaystyle x^2=(ai)^2$

$\displaystyle x=\pm ai$

3. ## Re: imaginary numbers x^2 + a^2 = 0 question

Originally Posted by ineedmathshelp
my edexcel further maths book says i need to solve this equation giving the answer in terms of the positive number a.
x^2 + a^2 = 0.
the answer in the back of the text book is x = + or - ai
but i dont see how i can get this answer
For every real number $\displaystyle a$ it is true that $\displaystyle a^2\ge 0~.$
Therefore, there is no real solution for $\displaystyle x^2+a^2=0$.
Thus concept of complex numbers was developed. We introduce the number $\displaystyle i$ to the number system.
$\displaystyle i$ is a solution to $\displaystyle x^2+1=0$. So $\displaystyle i^2=-1$.
$\displaystyle (ai)^2+a^2=a^2(i)^2+a=-a^2+a^2=0~.$