Absolutely! Let's complete the square step by step for the given quadratic expression:
### Step 1: Identify the Coefficient of [tex]\( x \)[/tex]
First, we identify the coefficient of [tex]\( x \)[/tex] in the quadratic expression:
[tex]\[ x^2 + 20x \][/tex]
The coefficient of [tex]\( x \)[/tex] is 20.
### Step 2: Calculate the Missing Term
To complete the square, we need to add and subtract a specific term that makes the expression a perfect square trinomial. This term is calculated using the formula [tex]\( \left( \frac{b}{2} \right)^2 \)[/tex], where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex].
Here, [tex]\( b \)[/tex] is 20. Thus, the missing term is:
[tex]\[ \left( \frac{20}{2} \right)^2 = 10^2 = 100 \][/tex]
So, the missing term to add to our original quadratic expression is 100.
### Step 3: Write the Perfect Square Trinomial
Now we add the missing term to the quadratic expression:
[tex]\[ x^2 + 20x + 100 \][/tex]
### Step 4: Factor the Trinomial
The expression [tex]\( x^2 + 20x + 100 \)[/tex] is a perfect square trinomial, which can be factored into:
[tex]\[ x^2 + 20x + 100 = (x + 10)^2 \][/tex]
### Final Answer
Thus, the completed expression and its corresponding factorization are:
[tex]\[ x^2 + 20x + 100 = (x + 10)^2 \][/tex]
In summary:
[tex]\[ x^2 + 20x + \boxed{100} = (\boxed{x + 10})^2 \][/tex]
