Let's solve the equation [tex]\( x^2 = 1 \)[/tex].
1. Start with the given equation:
[tex]\[ x^2 = 1 \][/tex]
2. To find the values of [tex]\( x \)[/tex], we need to take the square root of both sides of the equation. Remember, taking the square root yields two possible solutions: a positive and a negative root. Thus:
[tex]\[ x = \pm\sqrt{1} \][/tex]
3. Calculate the square root of 1:
[tex]\[ \sqrt{1} = 1 \][/tex]
4. Therefore, the solutions are:
[tex]\[ x = 1 \quad \text{and} \quad x = -1 \][/tex]
To summarize, the equation [tex]\( x^2 = 1 \)[/tex] has two solutions: [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].
Let's now look at the given answer choices:
a) [tex]\( \sqrt{1} \)[/tex]
b) 1
c) [tex]\( \sin^2 \theta + \cos^2 \theta \)[/tex]
Choice a) [tex]\( \sqrt{1} \)[/tex] is partly correct, but it does not account for the negative root.
Choice b) 1 is only one of the solutions and doesn’t cover the negative root.
Choice c) [tex]\( \sin^2 \theta + \cos^2 \theta \)[/tex] equals 1, but it still doesn't address the second solution of [tex]\(-1\)[/tex].
Therefore, the complete and accurate solutions to the equation [tex]\( x^2 = 1 \)[/tex] are [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex], which exactly match the numerical result from our calculations.
