Let's solve the given equation \((1 - \sqrt{3})^2 + (1 - \sqrt{3})^2 = c^2\) step-by-step.
1. Calculate \((1 - \sqrt{3})^2\):
To start, we need to expand the binomial expression \((1 - \sqrt{3})^2\).
[tex]\[
(1 - \sqrt{3})^2 = (1 - \sqrt{3})(1 - \sqrt{3})
\][/tex]
Let's expand this expression:
[tex]\[
(1 - \sqrt{3})^2 = 1^2 - 2 \cdot 1 \cdot \sqrt{3} + (\sqrt{3})^2
\][/tex]
Simplifying each term:
[tex]\[
1^2 = 1
\][/tex]
[tex]\[
-2 \cdot 1 \cdot \sqrt{3} = -2\sqrt{3}
\][/tex]
[tex]\[
(\sqrt{3})^2 = 3
\][/tex]
Putting it all together:
[tex]\[
(1 - \sqrt{3})^2 = 1 - 2\sqrt{3} + 3
\][/tex]
Simplifying further:
[tex]\[
(1 - \sqrt{3})^2 = 1 + 3 - 2\sqrt{3} = 4 - 2\sqrt{3}
\][/tex]
However, it appears from the given values that the precise value of \((1 - \sqrt{3})^2\) is approximately 0.5358983848622453.
2. Calculate \(c^2\):
We need to add the two expressions since \((1 - \sqrt{3})^2\) appears twice in our given equation:
[tex]\[
(1 - \sqrt{3})^2 + (1 - \sqrt{3})^2 = c^2
\][/tex]
We know from the given values that:
[tex]\[
(1 - \sqrt{3})^2 \approx 0.5358983848622453
\][/tex]
So:
[tex]\[
c^2 = 0.5358983848622453 + 0.5358983848622453
\][/tex]
Adding these together:
[tex]\[
c^2 \approx 1.0717967697244906
\][/tex]
3. Calculate \(c\):
To find \(c\), we take the square root of \(c^2\):
[tex]\[
c = \sqrt{1.0717967697244906}
\][/tex]
From the given values, we know this is approximately:
[tex]\[
c \approx 1.035276180410083
\][/tex]
So, to summarize:
- \((1 - \sqrt{3})^2 \approx 0.5358983848622453\)
- Adding these together: \( (1 - \sqrt{3})^2 + (1 - \sqrt{3})^2 \approx 1.0717967697244906\)
- Taking the square root to find \(c\): \(c \approx 1.035276180410083\)
Therefore, the solution to the given equation is approximately [tex]\(c \approx 1.035276180410083\)[/tex].
