Question

solve by completing the square,what value should be added to each side of the equation?
x^2 + 14x = -7

Answer 1

1/2 the middle term squared
so

14 is the middle term number
14/2 = 7
7^2 = 49

Answer 2

then notice that x^2 + 14x + 49
is factorable

Answer 3

Don't even have to notice that — if you've added the square of half the coefficient of the x term, then the left hand side becomes (x+14/2)^2 or (x+7)^2

Answer 4

though yes, hopefully you do notice it :-)

Answer 5

i was only pointing it out, because if you did the process incorrectly, your result would not be factorable

since it was factorable, must be the right process lol

Answer 6

so if I add (x+7)^2 to each side I can square it?

Answer 7

noo you dont need to make a binomial

look at it this way

x^2 + 14x + C

you want to find some number C, so that you can factor the whole thing

Answer 8

to do that, just create the value of C from the other numbers

namely, the coefficient of your middle term, which is 14

Answer 9

if you divide 14 by 2 (take half of it), then square that number, you will create a number which can be factored

Answer 10

the reason you take half, is because you are going to factor into 2 terms, and those two terms need to add up to 14

if you take half of 14, you get 7 and 7, which adds up perfectly to 14

the reason you square 7, is because you are factoring, and need a number which has a perfect square root (49, in this case)

Answer 11

mmm, no, you're just rewriting the left hand side as a perfect square. \[(x+14/2)^2 = (x+7)^2 = x^2+14x+49\]You had to add something to both sides so that the left side fit into that form. Now you have\[(x+7)^2 = -7 + 7^2\]\[(x+7)^2=42\]You can take the square root of both sides and find \(x\):
\[\sqrt{(x+7)^2} = \sqrt{42}\]\[\pm(x+7) = \sqrt{42}\]You have to solve with both the positive and negative square roots to get both solutions, so that is shorthand for
\[(x+7) = \sqrt{42}\]\[-(x+7) = \sqrt{42}\]