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

Answer:

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

Step-by-step 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.