Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

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Step-by-Step Solution

Answer

The simplified expression is \(|x+2|\), which equals \(x+2\) for \(x\geq-2\) and \(-x-2\) for \(x<-2\).

Step 1: Identify the Expression

The expression given is \( (x^2 + 4x + 4)^{1/2} \). This is under a square root sign, \( \sqrt{x^2 + 4x + 4} \).

Step 2: Recognize the Perfect Square

Notice that \( x^2 + 4x + 4 \) can be rewritten as \( (x + 2)^2 \). This is because \( (x + 2)^2 = x^2 + 2\cdot x\cdot 2 + 2^2 = x^2 + 4x + 4 \).

Step 3: Simplify the Expression

Substitute \( (x + 2)^2 \) into the expression: \( \sqrt{(x + 2)^2} \). The square root of a square expression is the absolute value: \( |x + 2| \).

Step 4: Determine the Absolute Value

The absolute value \( |x + 2| \) implies that it could be either \( x + 2 \) or \( -(x + 2) \) based on the value of \( x \). Specifically, \( |x + 2| = x + 2 \) when \( x \geq -2 \) and \( |x + 2| = -(x + 2) \) when \( x < -2 \).

Step 5: Conclude the Simplification

The simplified expression for \( \sqrt{x^2 + 4x + 4} \) is \( |x + 2| \). Thus, this is expressed as \( x + 2 \) for \( x \geq -2 \) or \( -x - 2 \) for \( x < -2 \).

Key Concepts

Square Root

The square root is an essential concept in algebra and mathematics overall. When we discuss the square root of a number, we are essentially asking, "What number, when multiplied by itself, will give me the original number?" The square root is denoted using a radical symbol, like this: \( \sqrt{ } \).
For instance, the square root of 9 is 3 because \( 3 \times 3 = 9 \).
Square roots are crucial in solving quadratic equations and simplifying expressions.

• To find the square root of a perfect square, simply determine the number that, squared, results in the given number.
• Always remember that square roots yield both positive and negative roots, represented as \( \pm \). For example, \( \sqrt{4} = \pm 2 \) because both \( 2 \times 2 \) and \( -2 \times -2 \) equal 4.

Understanding square roots can greatly assist in simplifying algebraic expressions, just like in the problem we're reviewing here, where recognizing a perfect square led us to a simple solution.

Perfect Square

A perfect square is a number that can be expressed as the square of an integer. Recognizing a perfect square can vastly simplify algebraic equations and expressions.
For example, \( 25 \) is a perfect square because it can be written as \( 5^2 \).
When tackling algebra problems, spotting and factoring perfect squares is a powerful tool.

• The term \((x+2)^2\) is a perfect square because it equals \(x^2 + 4x + 4\) when expanded.
• Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), you can rapidly identify and factor expressions.

In practice, watch for patterns to recognize perfect squares, such as examining quadratic expressions. By doing this, like with our exercise, you can replace more complex structures with simpler terms, facilitating other operations like square roots.

Absolute Value

Absolute value represents the distance of a number from zero on the numerical line, irrespective of direction.
It is always a non-negative value, indicated by two straight lines on either side of the number, like \( |x| \).
For example, \(|-5| = 5\) and \(|5| = 5\), since both are five units away from zero.

• In the given exercise, \(|x + 2|\) tells us the value of \(x + 2\) regardless if \(x\) is positive or negative.
• Mathematically, \(|x| = x\) when \(x\geq 0\) and \(|x| = -x\) when \(x < 0\).
• This is why in our exercise, \(|x + 2| = x + 2\) for \(x\geq -2\) and \(-x - 2\) for \(x < -2\).

The absolute value is essential in expressions because it affects the output's sign, ensuring you always obtain a non-negative solution. Understanding absolute values lets you correctly simplify and interpret solutions in algebra.