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 $8$.
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