Let's solve the equation [tex]\( x^2 + 10x + \ldots \)[/tex] to express it in the form [tex]\( (x + a)^2 \)[/tex].
1. Start with the given quadratic expression [tex]\( x^2 + 10x \)[/tex].
2. To complete the square, we need to add and subtract a specific value that makes the expression a perfect square trinomial.
3. The general form of a perfect square trinomial is [tex]\( (x + a)^2 = x^2 + 2ax + a^2 \)[/tex].
4. Identify the coefficient of [tex]\( x \)[/tex], which is 10 in this case, and divide it by 2:
[tex]\[
\frac{10}{2} = 5
\][/tex]
5. Square the result obtained in step 4:
[tex]\[
5^2 = 25
\][/tex]
6. Thus, the value we need to add to complete the square is 25. This transforms our original expression into a perfect square trinomial:
[tex]\[
x^2 + 10x + 25
\][/tex]
7. This trinomial can be factored as:
[tex]\[
(x + 5)^2
\][/tex]
Therefore, the complete equation should be:
[tex]\[
x^2 + 10x + 25 = (x + 5)^2
\][/tex]
Given the options:
A. [tex]\( 10 ; x + 10 \)[/tex]
B. [tex]\( 25 ; x + 5 \)[/tex]
C. [tex]\( 10 ; x - 10 \)[/tex]
D. [tex]\( 25 ; x - 5 \)[/tex]
The correct answer is:
[tex]\[ \text{B. } 25 ; x + 5 \][/tex]
So, the completed equation is:
[tex]\[ x^2 + 10x + 25 = (x + 5)^2 \][/tex]
Therefore, [tex]\( 25 ; x + 5 \)[/tex] is the correct answer.
