Question

Prove that 1 and −1 are the only solutions to the equation x^2 = 1.

Let x = a + bi be a complex number so that x^2 = 1.
a. Substitute a + bi for x in the equation x^2 = 1.
b. Rewrite both sides in standard form for a complex number.
c. Equate the real parts on each side of the equation, and equate the imaginary parts on each side of the
equation.
d. Solve for a and b, and find the solutions for x = a + bi.

Answer 1

1 and −1 are the only solutions to the equation x^2 = 1.

Explanation:

We shall proceed as he suggests

`x=a+bi`

Given
`x^(2)=1`

substitute a+bi in x, we get

`(a+bi)^(2)=1`

Rewriting the both sides in standard form for a complex number

`(a^(2)-b^(2))+2abi=1+0i`

Equating the real parts on each side of the equation, and equating the imaginary parts on each side of the equation.

`a^(2)-b^(2)=1` and
`2ab=0`

So either a=0 or b=0. If a=0 then

`0^(2)-b^(2)=1`

`b^(2)=-1` . has no real solution.

If b=0 then

`a^(2)-0^(2)=1`

`a^(2)=1`

`a^(2)-1^(2)=0`

`(a-1)(a+1)=0`

`a=1` . or
`a=-1`

Hence proved.