Problem 9 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{x^{2}+4 x+4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(\sqrt{x^2 + 4x + 4}\). Answer: \(x + 2\).

Step 1: Analyzing the quadratic polynomial

Inside the square root, we have a quadratic expression: \(x^{2}+4x+4\). Let's see if we can factor it further.

Step 2: Factoring the quadratic polynomial

In this case, we notice that the quadratic polynomial is a perfect square trinomial which has the form \((x+a)^2\). Therefore, the factored expression will be: \((x+2)^2\), where \(a = 2\).

Step 3: Simplify the expression

Now, we can rewrite and simplify the given expression as follows: $$ \sqrt{x^{2}+4 x+4} = \sqrt{(x+2)^2} $$ The square root and the square cancel each other out, so the simplified expression is: $$ x+2 $$

Key Concepts

Factoring Quadratic Polynomials

Factoring quadratic polynomials is a foundational skill in algebra. It involves finding two binomials that, when multiplied together, give back the original quadratic polynomial.

The general form of a quadratic polynomial is \( ax^2 + bx + c \), where \( a \) , \( b \) and \( c \) are constants. When factoring, one looks for two numbers that both add up to \( b \) and multiply to \( ac \).

For example, consider the quadratic expression \( x^2 + 4x + 4 \). To factor this, we need to find two numbers that add up to 4 (the coefficient of the middle term) and also multiply to 4 (the constant term). These numbers are both 2, providing the factors \( (x+2)(x+2) \) or \( (x+2)^2 \).

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic polynomial that can be expressed as the square of a binomial. In other words, it's the result of squaring a binomial.

The standard form of a perfect square trinomial is \( (x+a)^2 \), which expands to \( x^2 + 2ax + a^2 \). Notice that the first and last terms are perfect squares, and the middle term is twice the product of the roots of those squares.

For the quadratic expression we're working with, \( x^2 + 4x + 4 \), we can see it fits this form with \( a = 2 \). Therefore, it's a perfect square trinomial since \( (x+2)^2 = x^2 + 2*2*x + 2^2 \). Recognizing this pattern is key to simplification.

Square Root Simplification

Square root simplification involves finding the non-negative root of a perfect square number or expression. The square root, represented by the root symbol \( \sqrt{} \), is the inverse operation of squaring.

To simplify the square root of a perfect square trinomial, such as \( \sqrt{(x+2)^2} \), we take the square root of both the binomial and the squared sign. Since \( (x+2)^2 \), is a perfect square, its square root will simply be the binomial itself, which is \( x+2 \).

This process is direct and follows from the property that \( \sqrt{a^2} = a \) when \( a \) is non-negative. Thus, \( \sqrt{(x+2)^2} \), simplifies directly to \( x+2 \) without any extra steps or complications, provided that \( x+2 \) is non-negative, which is implied in this context.