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3 Responses
You’re super close! The only hiccup is that when you go from x^2 + 6x to (x + 3)^2, you’ve secretly added a 9, because (x + 3)^2 expands to x^2 + 6x + 9. So if you write (x + 3)^2 + 5, you’ve actually created x^2 + 6x + 14. That’s 9 too big.
Here’s the clean “complete the square” path:
1) Start with x^2 + 6x + 5.
2) Take half of the middle coefficient: half of 6 is 3. Square it: 3^2 = 9. That’s the number that completes the square.
3) Add and subtract that 9 so you don’t change the value:
x^2 + 6x + 5 = (x^2 + 6x + 9) + 5 − 9.
4) Now the first three terms are a perfect square:
x^2 + 6x + 9 = (x + 3)^2.
5) So the whole thing is:
(x + 3)^2 + 5 − 9 = (x + 3)^2 − 4.
That’s the completed-square form: (x + 3)^2 − 4.
Quick check: expand (x + 3)^2 − 4 = x^2 + 6x + 9 − 4 = x^2 + 6x + 5. Perfect.
A tiny analogy: Think of it like balancing a seesaw. To turn x^2 + 6x into a perfect square, you “place” 9 units on one side. To keep the seesaw level, you have to “remove” 9 somewhere else. You added 9 inside the square, so you subtract 9 outside; combined with the +5, that leaves −4.
General pattern to remember:
x^2 + bx + c = (x + b/2)^2 − (b/2)^2 + c.
Here b = 6, so (b/2)^2 = 9, and you get (x + 3)^2 − 9 + 5 = (x + 3)^2 − 4.
You’ve got the right instinct-just don’t forget to subtract the extra 9 you slipped in when you made the square!
You’re so close-just one tiny bookkeeping step! Completing the square means you add and subtract the same perfect square so you don’t change the value. Here, for x^2 + 6x + 5, take half of 6 (that’s 3), square it to get 9, then add and subtract 9: x^2 + 6x + 5 = (x^2 + 6x + 9) + 5 − 9 = (x + 3)^2 − 4. Your rewrite (x + 3)^2 + 5 sneaks in an extra +9 without paying it back-expanding it gives x^2 + 6x + 14, which is 9 too large. Quick check: expand (x + 3)^2 − 4 → x^2 + 6x + 9 − 4 = x^2 + 6x + 5, perfect. And as a little bonus, this is the vertex form, so the parabola’s vertex is at (−3, −4)-neat!
Half the 6 is 3, so add and subtract 9: x^2 + 6x + 5 = (x^2 + 6x + 9) + 5 − 9 = (x+3)^2 − 4-the +5 version forgets to cancel the extra +9 you just smuggled in. Want to try the same trick on x^2 − 8x + 1?