Solving Expression Value for x = sqrt(3)

We need to calculate the value of the expression $(x + 1)^2 + (x - 1)^2$ given that $x = \sqrt{3}$.

Expression Simplification

First, expand the two squared terms using the binomial expansion formulas:

• $(a+b)^2 = a^2 + 2ab + b^2 \implies (x + 1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1$
• $(a-b)^2 = a^2 - 2ab + b^2 \implies (x - 1)^2 = x^2 - 2(x)(1) + 1^2 = x^2 - 2x + 1$

Now, add the expanded forms together:

$(x^2 + 2x + 1) + (x^2 - 2x + 1)$

Combine the like terms:

$x^2 + x^2 + 2x - 2x + 1 + 1 = 2x^2 + 2$

Substitution and Final Calculation

Substitute the given value $x = \sqrt{3}$ into the simplified expression $2x^2 + 2$:

$2(\sqrt{3})^2 + 2$

Evaluate the square root and multiply:

$2(3) + 2$

$6 + 2 = 8$

The value of the expression $(x + 1)^2 + (x - 1)^2$ when $x = \sqrt{3}$ is $8$.