To complete the square for the expression [tex]\(x^2 + 14x - 10\)[/tex], we need to follow a systematic method. Here, we want to form a perfect square trinomial from the terms involving [tex]\(x\)[/tex]. Let's break it down step-by-step:
### Step-by-Step Solution:
1. Identify the coefficient of the [tex]\(x\)[/tex] term:
- The expression given is [tex]\(x^2 + 14x - 10\)[/tex].
- The coefficient of the [tex]\(x\)[/tex] term is [tex]\(14\)[/tex].
2. Take half of the coefficient of [tex]\(x\)[/tex] and square it:
- Half of [tex]\(14\)[/tex] is [tex]\(7\)[/tex].
- Squaring [tex]\(7\)[/tex] gives [tex]\(49\)[/tex] (since [tex]\(7^2 = 49\)[/tex]).
3. Form a perfect square trinomial:
- We need to add and subtract [tex]\(49\)[/tex] within our expression to transform it into a perfect square trinomial.
- Thus, the expression becomes:
[tex]\[
x^2 + 14x - 10 + 49 - 49
\][/tex]
- Rearrange the terms:
[tex]\[
x^2 + 14x + 49 - 10 - 49
\][/tex]
4. Rewrite the expression in the form of a perfect square:
- The perfect square trinomial we formed is [tex]\(x^2 + 14x + 49\)[/tex].
- This trinomial can be written as:
[tex]\[
(x + 7)^2
\][/tex]
- So, the expression becomes:
[tex]\[
(x + 7)^2 - 59
\][/tex]
### Conclusion:
The number you need to add to the expression [tex]\(x^2 + 14x - 10\)[/tex] to make it a perfect square trinomial is [tex]\(\boxed{49}\)[/tex].
