Complete the equation: [tex]$x^2+10 x+\ldots=()^2$[/tex]

A. [tex]$10 ; x+10$[/tex]

B. [tex][tex]$25 ; x+5$[/tex][/tex]

C. [tex]$10 ; x-10$[/tex]

D. [tex]$25 ; x-5$[/tex]



Answer :

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.