What will come in place of ? to satisfy the equation : $(\sqrt{5}-1)^2 = ?-2\sqrt{5}$
Solve Algebraic Equation: Find ? in $(\sqrt{5}-1)^2$
The problem requires finding the value that replaces the question mark '?' in the given equation.
Step 1: Expand the Left Side Expression
We need to expand the term $(\sqrt{5}-1)^2$. Using the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$, where $a = \sqrt{5}$ and $b = 1$.
$(\sqrt{5}-1)^2 = (\sqrt{5})^2 - 2(\sqrt{5})(1) + (1)^2$
Calculating the terms:
- $(\sqrt{5})^2 = 5$
- $2(\sqrt{5})(1) = 2\sqrt{5}$
- $(1)^2 = 1$
Substituting these back into the expansion:
$(\sqrt{5}-1)^2 = 5 - 2\sqrt{5} + 1$
Combine the constant terms:
$(\sqrt{5}-1)^2 = 6 - 2\sqrt{5}$
Step 2: Substitute and Solve for ?
The original equation is $(\sqrt{5}-1)^2 = ?-2\sqrt{5}$.
Now, substitute the expanded form of the left side:
$6 - 2\sqrt{5} = ?-2\sqrt{5}$
To find the value of '?', isolate it by adding $2\sqrt{5}$ to both sides of the equation:
$6 - 2\sqrt{5} + 2\sqrt{5} = ? - 2\sqrt{5} + 2\sqrt{5}$
This simplifies to:
$6 = ?$
Conclusion
The value that comes in place of '?' is 6.
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