Question

Factor the perfect squares. $$ x^{2}+4 x+4 $$

Short answer

\(x^2 + 4x + 4 = (x + 2)^2\).

Step-by-step solution

Identify the trinomial form

The given expression is a trinomial in the form of \[ax^2 + bx + c\]. In this case, \(a = 1\), \(b = 4\), and \(c = 4\).

Express the middle term

Check if the middle term \(4x\) can be written as \(2 \times x \times 2\). Here, \(4x = 2 \times x \times 2\).

Perfect square transformation

Rewrite the trinomial using the perfect square formula \[(a + b)^2 = a^2 + 2ab + b^2\]. Here, \(a = x\) and \(b = 2\), so it becomes \((x + 2)^2\).

Write the factorized form

Therefore, \(x^2 + 4x + 4\) can be written as \((x + 2)^2\).

Key concepts

Trinomial Form

When you deal with expressions like \[x^2 + 4x + 4\], you handle a special polynomial known as a trinomial. A trinomial is simply a polynomial with three terms. The general form of such a trinomial is \[ax^2 + bx + c\], where \(a\), \(b\), and \(c\) are constants.
In our example, we can see directly that \(a = 1\), \(b = 4\), and \(c = 4\). Recognizing this form is crucial because it allows us to apply specialized methods to simplify or factorize the expression.
Always start by ensuring that the expression fits this standard form. This establishes the basics for any operations you will carry out.

Perfect Square Formula

The perfect square formula is a useful tool when working with trinomials. The formula is given by \[(a + b)^2 = a^2 + 2ab + b^2\]. This formula tells us how a squared binomial looks once expanded.
To use this for trinomial factorization, you should identify \(a\) and \(b\) in the given expression. For \[x^2 + 4x + 4\], observe the middle term \(4x\). We can write \(4x\) as \(2 \times x \times 2\), with \(a = x\) and \(b = 2\).
Now, fit these values into the perfect square formula:
The given expression \[x^2 + 4x + 4\] matches \[a^2 + 2ab + b^2\], implying that the expression can be rewritten as \[(x + 2)^2\]. This rewritten form reveals that the original trinomial is a perfect square.

Factorization Steps

Let's walk through the steps to factorize a given perfect square trinomial, \[x^2 + 4x + 4\].
Step 1: Identify Trinomial Form

• Confirm that the expression is \[ax^2 + bx + c\].
• In our example, \(a = 1\), \(b = 4\), \(c = 4\).

Step 2: Check Middle Term

• Verify if the middle term \(4x\) can be expressed as \(2 \times x \times 2\).
• This shows \(a = x\) and \(b = 2\).

Step 3: Apply Perfect Square Formula

• Use the formula \[(a + b)^2 = a^2 + 2ab + b^2\].
• With \(a = x\) and \(b = 2\), rewrite as \[(x + 2)^2\].

Step 4: Final Factorized Form

• The trinomial \[x^2 + 4x + 4\] is now factorized as \[(x + 2)^2\].

These steps break down the process into manageable actions, helping you transform and factorize perfect square trinomials with ease.