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 $8$.
Similar Questions
If \(x+\dfrac{1}{x}=2\sqrt{3}\), then find the value of \(x^3+\dfrac{1}{x^3}\).
Important Questions from Identities
The coefficient of y in the expansion of (2y – 5) 3, is:
If x + y = 2 and \(\frac{1}{x}+\frac{1}{y}=\frac{18}{5}\) , then the value of (x 3+ y 3) is:
If x - y = 11 and \(\rm \frac{1}{x} - \frac{1}{y} = \frac{11}{24}\) then the value of x 3 - y 3 + x 2y 2 ?
If 2x 2- 8x - 1 = 0, then what is the value of \(\rm 8x^3 - \frac{1}{x^3}\) ?
If \(\rm x+ \frac{1}{x} = 4,\) then the value of \(\rm x^5 + \frac{1}{x^5}\) is:
