The answer is 6, here's why.
So our function is f(x) = x^2 - 12x + 7.
To complete the square, because there is no greatest common factor first we separate the variable terms from the constant by putting them in parenthesis.
x^2 - 12x + 7 becomes
(x^2 - 12x) + 7.
Next, we take the coefficient for the second term in parenthesis, half it, and then square the result.
Half of 12 is 6, and 6^2 = 36. We now add this inside the parenthesis to get
(x^2 - 12x + 36) + 7
Now we have to subtract 36 outside the parenthesis, because we added 36 inside.
(x^2 - 12x + 36) + 7 - 36
(x^2 - 12x + 36) - 29.
Then we factor the trinomial inside the parenthesis.
x^2 - 12x + 36
Find the pair of numbers that multiply to get the last term, and add to get the middle term.
In this case, our answer is -6, because -6*-6 = 36, and -6 + -6 = -12.
Replace the middle term with this pair of numbers.
x^2 - 6x - 6x + 36
Now, separate the polynomial into two groups and factor out the GCF (Greatest Common Factor) of each term.
We have x^2 - 6x and - 6x + 36
The GCF of x^2 - 6x is x.
The GCF of -6x + 36 is -6.
Factor the GCFs out and get
x(x - 6) -6(x - 6)
The GCF of x(x - 6) -6(x - 6) as you can see is (x - 6). Factor that out to get
(x - 6)(x - 6), or (x - 6)^2.
Therefore, (x^2 - 12x + 36) - 29 simplified is (x - 6)^2 - 29.
f(x) = x^2 - 12x + 7 in vertex form is (x - 6)^2 - 29, and the value of a, as you can see, is 6.
Let me know if this answers your question, and if you need help understanding anything. I'll try to explain the best I can if so.
Hope this helps!
