Simplify square root of 2a^2+4a+2
The question asks to perform a simplification on a mathematical expression that involves a square root. Specifically, the expression under the square root is a quadratic expression given by 2a^2+4a+2. The task is to rewrite this expression in a simpler or more concise form if possible, which typically involves factoring or using other algebraic techniques to simplify the expression inside the square root.
$\sqrt{2 a^{2} + 4 a + 2}$
Answer
Solution:
Step:1
Extract the factor of $2$ from the expression $2a^2 + 4a + 2$.
Step:1.1
Take $2$ out from $2a^2$. $\sqrt{2(a^2) + 4a + 2}$
Step:1.2
Remove $2$ from $4a$. $\sqrt{2(a^2) + 2(2a) + 2}$
Step:1.3
Extract $2$ from the constant term $2$. $\sqrt{2(a^2) + 2(2a) + 2(1)}$
Step:1.4
Factor out $2$ from the terms $2a^2 + 4a$. $\sqrt{2(a^2 + 2a) + 2(1)}$
Step:1.5
Complete the extraction of $2$ from the entire expression. $\sqrt{2(a^2 + 2a + 1)}$
Step:2
Apply the perfect square factoring technique.
Step:2.1
Express $1$ as $1^2$. $\sqrt{2(a^2 + 2a + 1^2)}$
Step:2.2
Verify that the middle term is twice the product of the square roots of the first and last terms. $2a = 2 \cdot a \cdot 1$
Step:2.3
Reformulate the quadratic expression. $\sqrt{2(a^2 + 2 \cdot a \cdot 1 + 1^2)}$
Step:2.4
Factor the trinomial using the formula $(a + b)^2 = a^2 + 2ab + b^2$, where $a = a$ and $b = 1$. $\sqrt{2((a + 1)^2)}$
Step:3
Rearrange $2((a + 1)^2)$ to $(a + 1)^2 \cdot 2$.
Step:3.1
Switch the positions of $2$ and $(a + 1)^2$. $\sqrt{((a + 1)^2 \cdot 2)}$
Step:3.2
Represent $1$ as $1^2$. $\sqrt{((a + 1^2)^2 \cdot 2)}$
Step:4
Extract terms from under the square root. $(a + 1^2)\sqrt{2}$
Step:5
Recognize that any number raised to the power of one remains the same. $(a + 1)\sqrt{2}$
Knowledge Notes:
Factoring out a common term: This involves taking out a common factor from all terms in an expression. In this case, we factor out a $2$ from each term in the quadratic expression.
Perfect square trinomial: A trinomial of the form $a^2 + 2ab + b^2$ is a perfect square because it can be factored into $(a + b)^2$. In the problem, we recognize that the expression inside the square root is a perfect square trinomial.
Square root of a product: The square root of a product, $\sqrt{xy}$, can be expressed as the product of the square roots, $\sqrt{x}\sqrt{y}$, provided that $x$ and $y$ are non-negative.
Simplifying square roots: When a term under a square root is a perfect square, it can be taken out of the square root as the base of the square. For example, $\sqrt{a^2} = a$.
Exponent rules: Any number raised to the power of one is itself, and raising a number to the power of two squares the number. These rules are used to simplify expressions within the square root.
